#6676455

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6583651);