#126045

order	512
nilpotency class	4
isoclinism	isoclinic class of #125705
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×#95	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂×#95 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

538 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,4,7)(3,5,8,6)(9,10)(11,13)(12,15)(14,16)(17,18,21,25)(19,23,26,32)(20,22,29,24)(27,30,28,31);(2,5)(3,8)(6,7)(18,22,25,24)(19,26)(23,31,32,30)(27,28);(10,13)(11,16)(14,15)(17,19,21,26)(18,23,25,32)(20,28,29,27)(22,30,24,31)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,c^b=ci,d^b=dfi,e^b=egi,d^c=dg,e^c=eh,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33241);