#72343625

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂³×#106
prime radical	C₂⁵	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4017 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

(5,6,9,15,11,14,28,17)(7,18,21,46,20,45,26,43)(8,13,24,38,23,37,52,40)(10,29,27,16,31,42,51,41)(12,32,35,22,34,30,36,50)(19,44,48,65,47,66,54,64)(25,56,55,39,53,63,68,62)(33,58,59,49,61,57,60,67);(2,4)(5,7,11,20)(6,12)(8,19,23,47)(9,26,28,21)(10,30,31,32)(13,33)(14,34)(15,36)(16,43)(17,35)(18,29)(22,51,50,27)(24,54,52,48)(25,57,53,58)(37,61)(38,60)(39,64)(40,59)(41,46)(42,45)(44,56)(49,68,67,55)(62,65)(63,66);(1,2,3,4)(5,8)(6,13)(7,19)(9,24)(10,25)(11,23)(12,33)(14,37)(15,38)(16,39)(17,40)(18,44)(20,47)(21,48)(22,49)(26,54)(27,55)(28,52)(29,56)(30,57)(31,53)(32,58)(34,61)(35,59)(36,60)(41,62)(42,63)(43,64)(45,66)(46,65)(50,67)(51,68);(7,20)(8,23)(12,34)(13,37)(18,45)(21,26)(22,50)(24,52)(25,53)(30,32)(35,36)(38,40)(39,62)(43,46)(55,68)(56,63);(6,14)(8,23)(12,34)(15,17)(16,41)(18,45)(19,47)(24,52)(25,53)(29,42)(35,36)(43,46)(48,54)(49,67)(55,68)(57,58)

Polycyclic presentation:

2^10|a^2=f,b^2=fi,c^2=g,f^2=j,h^2=i,i^2=j,b^a=bh,c^b=cg,d^b=dj,d^c=dj,e^a=ej,e^c=ej,f^b=fj,h^a=hi,h^b=hi,i^a=ij,i^b=ij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343634, #72343754, #72343764
maximal normal subgroup of	#72343634, #72343754, #72343764

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.6*F.9);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);