#93467

order	512
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#342
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂²×#117	C₂²
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×#117 ▷ C₂⁴ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=h,e^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fi,c^b=cg,d^b=dfi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 663);