#232413

order	512
nilpotency class	3
isoclinism	isoclinic class of #232393
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

1384 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1384 conjugacy classes of subgroups — too many to draw. Only its 266 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)(5,6,10,7)(8,11,9,12)(13,14,19,16)(15,22,25,28)(17,27,18,21)(20,24,26,23);(6,11)(7,12)(8,9)(13,15,19,25)(14,20,16,26)(17,24,18,23)(21,28,27,22);(2,4)(5,7,10,6)(8,12,9,11)(13,16,19,14)(15,22,25,28)(17,27,18,21)(20,23,26,24);(5,8,10,9)(6,12,7,11)(14,16)(20,26)(21,27)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139609);