#232371

order	512
nilpotency class	3
isoclinism	isoclinic class of #232351
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1304 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 258 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,14)(7,12,17,20)(8,13,9,11)(15,19,16,18)(21,22,27,24)(23,31,35,32)(25,30,26,29)(28,34,36,33);(6,11)(7,15)(8,9)(13,14)(16,17)(18,19)(21,23,27,35)(22,28,24,36)(25,34,26,33)(29,32,30,31);(2,4)(5,7,10,17)(6,12,14,20)(8,15,9,16)(11,18,13,19)(21,24,27,22)(23,32,35,31)(25,29,26,30)(28,33,36,34);(5,8,10,9)(6,13,14,11)(7,15,17,16)(12,19,20,18)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139567);