#6683061

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#32939	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2010 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)(13,14,17,19)(15,23,24,39)(16,28,30,21)(18,32,25,22)(20,34,33,27)(26,41,29,38)(31,37,40,43)(35,42,36,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(15,24)(21,28)(22,32)(26,29)(27,34)(37,43)(42,44);(2,4)(13,15,25,34,17,24,18,27)(14,20,32,39,19,33,22,23)(16,29,40,42,30,26,31,44)(21,36,43,38,28,35,37,41);(6,10)(9,11)(13,16)(14,21)(15,26)(17,30)(18,31)(19,28)(20,35)(22,37)(23,38)(24,29)(25,40)(27,42)(32,43)(33,36)(34,44)(39,41);(2,4)(7,12)(9,11)(16,30)(21,28)(26,29)(31,40)(35,36)(37,43)(38,41)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,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, 6590257);