#125053

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 294 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32249);