#93389

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂²
socle	C₂³	#340
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,27,24,29,23,38)(19,30,32,41,35,46,34,48)(20,28,36,43,37,44,22,26)(25,39,40,47,42,33,45,31);(2,5,7,6)(9,11)(10,13,14,15)(12,16)(17,19,22,33,23,34,37,47,24,35,36,39,21,32,20,31)(18,25,28,41,27,40,43,46,29,42,44,48,38,45,26,30)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=i,e^2=h,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,h^a=hi,c^b=cg

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 585);