#290136

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=efi,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

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, 197332);