#294527

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201723);