#6660331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636908
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	#37305	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6567527);