#125348

order	512
nilpotency class	4
isoclinism	isoclinic class of #125287
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32544);