#125116

order	512
nilpotency class	4
isoclinism	isoclinic class of #125094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

355 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fgi,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,d^a=df,c^b=ch,d^b=dg,e^b=efg,g^b=gi,h^b=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32312);