#291000

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	#24750	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

896 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)(9,10,14,11)(12,13,15,16)(17,18,22,25)(19,27,30,24)(20,21,26,23)(28,29,31,32);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,16,15,13)(17,19)(18,24)(20,31)(21,29)(22,30)(23,32)(25,27)(26,28);(9,12,13,11,14,15,16,10)(17,20,23,18,22,26,21,25)(19,28,29,24,30,31,32,27);(2,6)(5,7)(17,21,22,23)(18,20,25,26)(19,29,30,32)(24,31,27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,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, 198196);