#351427

order	512
nilpotency class	3
isoclinism	isoclinic class of #348358
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

584 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,18,21)(13,16,20,17)(19,22,24,23)(26,31)(27,29)(28,36)(32,39)(33,34)(35,37);(1,3,4,7)(2,5,6,8)(10,14)(15,21)(16,17)(22,23)(25,26,29,31,30,33,27,34)(28,32,36,38,37,39,35,40);(2,6)(5,8)(9,11,12,18)(10,15,14,21)(13,19,20,24)(16,22,17,23)(25,27,30,29)(26,31,33,34)(28,35,37,36)(32,38,39,40);(10,16)(13,20)(14,17)(15,22)(19,24)(21,23)(25,28,27,35,30,37,29,36)(26,32,34,40,33,39,31,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=efi,c^2=i,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ch,g^d=ghi

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, 258623);