#290980

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198176);