#255149

order	512
nilpotency class	3
isoclinism	isoclinic class of #245999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

938 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)(9,10,13,11)(12,14,16,15)(17,18,23,27)(19,20,28,31)(21,25,22,26)(24,29,32,30);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(17,19,23,28)(18,20,27,31)(21,24,22,32)(25,29,26,30);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(12,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32);(2,5)(6,8)(17,21,23,22)(18,25,27,26)(19,24,28,32)(20,29,31,30)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162345);