#6669129

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2087 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)(8,9)(11,12)(13,14,17,20)(15,23,26,19)(16,28,30,21)(18,22,32,37)(24,36,27,34)(25,35,43,41)(29,38,42,44)(31,40,33,39);(6,9)(7,11)(13,15,17,26)(14,19,20,23)(16,29,30,42)(18,24,32,27)(21,38,28,44)(22,34,37,36)(25,31,43,33)(35,39,41,40);(2,4)(5,7,8,11)(6,10,9,12)(15,24)(18,32)(19,34)(22,37)(23,36)(25,42)(26,27)(29,43)(31,33)(35,44)(38,41)(39,40);(6,9)(10,12)(14,20)(16,30)(19,23)(22,37)(25,43)(29,42)(31,33)(34,36);(13,16,17,30)(14,21,20,28)(15,25,26,43)(18,33,32,31)(19,35,23,41)(22,40,37,39)(24,42,27,29)(34,44,36,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,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, 6576325);