#352616

order	512
nilpotency class	3
isoclinism	isoclinic class of #352010
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259812);