#6810547

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717743);