#291147

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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

842 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)(9,10,15,11)(12,14,16,13)(17,18,20,22)(19,25,26,32)(21,29,27,24)(23,31,30,28);(1,3,4,8)(2,5,6,7)(9,11,15,10)(12,13,16,14)(18,22)(19,26)(21,27)(23,30);(9,12,14,11,15,16,13,10)(17,19,27,31,20,26,21,28)(18,23,29,32,22,30,24,25);(2,6)(5,7)(9,13,15,14)(10,12,11,16)(18,22)(23,30)(24,29)(25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fhi,d^2=i,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,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198343);