#6676883

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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
prime radical	#42800	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6584079);