#232315

order	512
nilpotency class	3
isoclinism	isoclinic class of #232313
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	C₂×#2656	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2044 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2044 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 139511);