#351120

order	512
nilpotency class	3
isoclinism	isoclinic class of #347758
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 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)(11,19)(12,15)(13,23)(14,18)(16,21)(17,20)(22,24)(25,26,28,31)(27,30,34,38)(29,32,36,33)(35,37,40,39);(2,5)(6,8)(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19)(25,27)(26,30)(28,34)(29,35)(31,38)(32,37)(33,39)(36,40);(1,3,4,7)(2,6,5,8)(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24)(26,31)(27,34)(32,33)(35,40);(26,32)(29,36)(30,37)(31,33)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,g^d=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258316);