#124804

order	512
nilpotency class	4
isoclinism	isoclinic class of #124681
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 302 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,5)(3,7)(6,8)(9,10,12,18,13,20,11,21,14,19,22,15,24,16,23,17);(1,2,4,6)(3,8,7,5)(10,15,19,18)(11,22)(12,23)(13,24)(16,17,20,21);(10,16,19,20)(11,23)(12,22)(15,17,18,21)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fghi,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32000);