#325500

order	512
nilpotency class	3
isoclinism	isoclinic class of #278091
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

549 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,14)(13,21)(15,20)(16,18)(19,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(1,3,5,4)(2,6,8,7)(10,14)(15,22)(16,21)(17,23)(25,27,29,35,28,34,36,33)(26,30,32,37,31,38,39,40);(1,3,5,4)(2,7,8,6)(9,11,18,24,12,19,13,20)(10,15,21,23,14,22,16,17)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(1,4,5,3)(2,7,8,6)(10,14)(15,22)(16,21)(17,23)(27,34)(30,38)(33,35)(37,40)

Polycyclic presentation:

2^9|b^2=egh,c^2=efi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=di,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, 232696);