#6817030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6,9,11)(7,12,8,10)(13,14,18,24)(15,23,19,20)(16,27,30,35)(17,31,34,22)(21,25,38,28)(26,36,33,40)(29,44,32,43)(37,42,39,41);(2,3)(5,6)(7,10)(8,12)(9,11)(13,15,18,19)(14,20,24,23)(16,28,30,25)(17,26,34,33)(21,27,38,35)(22,36,31,40)(29,42,32,41)(37,43,39,44);(1,2,4,3)(13,16,15,25,18,30,19,28)(14,21,20,35,24,38,23,27)(17,32,26,42,34,29,33,41)(22,39,36,43,31,37,40,44);(5,7,9,8)(6,10,11,12)(13,17,18,34)(14,22,24,31)(15,26,19,33)(16,29,30,32)(20,36,23,40)(21,37,38,39)(25,41,28,42)(27,43,35,44);(1,3,4,2)(5,8,9,7)(6,10,11,12)(13,15,18,19)(14,23,24,20)(16,25,30,28)(17,33,34,26)(21,27,38,35)(22,36,31,40)(29,42,32,41)(37,44,39,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724226);