#149308

order	512
nilpotency class	4
isoclinism	isoclinic class of #129405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

954 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=de,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,g^b=gh,g^c=gh,g^d=gi,g^e=gi

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, 56504);