#72343762

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

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

4449 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343764, #72343863, #72343880
maximal normal subgroup of	#72343764, #72343863, #72343880

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.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
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, 1, F.6*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);