#255510

order	512
nilpotency class	3
isoclinism	isoclinic class of #247047
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	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

1090 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)(3,6)(4,5)(7,8)(10,13)(11,14)(15,16)(17,18,20,24)(19,26,28,22)(21,23,30,25)(27,32,29,31);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19,20,28)(18,22,24,26)(21,27,30,29)(23,31,25,32);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(18,23)(19,27)(21,30)(24,25)(28,29)(31,32);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(18,24)(22,26)(23,25)(31,32)

Polycyclic presentation:

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

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, 162706);