#416615

order	512
nilpotency class	3
isoclinism	isoclinic class of #397950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35329	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi

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, 323811);