#6683099

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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1980 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)(13,14,18,22)(15,23,27,39)(16,20,28,35)(17,31,24,21)(19,32,34,26)(25,38,29,42)(30,44,40,37)(33,43,36,41);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(16,28)(20,35)(25,29)(30,40)(33,36)(37,44)(38,42)(41,43);(2,4)(13,15,24,32,18,27,17,26)(14,19,31,39,22,34,21,23)(16,29,40,41,28,25,30,43)(20,36,44,38,35,33,37,42);(6,10)(9,11)(13,16)(14,20)(15,25)(17,30)(18,28)(19,33)(21,37)(22,35)(23,38)(24,40)(26,41)(27,29)(31,44)(32,43)(34,36)(39,42);(2,4)(7,12)(9,11)(16,28)(20,35)(25,29)(30,40)(33,36)(37,44)(38,42)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6590295);