#350500

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257696);