#6658522

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632068
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#31835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,g^c=gi,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, 6565718);