#256124

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163320);