#496779

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

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

498 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)(10,14)(11,12)(17,18,21,26,22,23,29,30,20,25,28,32,19,24,27,31);(9,10,13,14)(11,16,12,15)(17,19,20,22)(18,23,25,24)(21,27,28,29)(26,30,32,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19,20,22)(18,24,25,23)(21,27,28,29)(26,31,32,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(18,25)(23,24)(26,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403975);