#497169

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

436 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:

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

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,d^a=de,c^b=ce,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404365);