#495433

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂³
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂²×C₈	C₂⁴
socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

564 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)(9,10)(11,14)(12,16)(13,15)(17,18,22,27,23,19,28,29,21,26,30,32,20,25,24,31);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,22,29,23,26,28,32,21,25,30,31,20,18,24,27);(9,12,13,11)(10,14,15,16);(1,3,4,7)(2,6,5,8)(17,20,21,23)(18,25,26,19)(22,24,30,28)(27,31,32,29)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402629);