#126165

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	C₂⁴	#120
LCS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂×#95 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 414 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)(10,13)(11,14)(15,16)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,28,32,29);(2,5)(3,8)(6,7)(9,10,12,13)(11,14,15,16)(17,19,22,30)(18,23,26,32)(20,29,21,28)(24,27,25,31);(9,11)(10,14)(12,15)(13,16)(17,20)(18,24,26,25)(19,28)(21,22)(23,31,32,27)(29,30)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=gi,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=df,e^b=egi,d^c=dg,e^c=ehi,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, 33361);