#350548

order	512
nilpotency class	3
isoclinism	isoclinic class of #346582
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257744);