#72343629

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂⁵	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4273 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fhi,c^2=gj,f^2=i,h^2=ij,i^2=j,b^a=bij,c^a=cj,c^b=cg,d^b=dj,d^c=dj,e^a=ej,e^c=ej,f^b=fj,h^a=hj,h^b=hj

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343634, #72343780, #72343863, …
maximal normal subgroup of	#72343634, #72343780, #72343863, …

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