#153016

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

240 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 76
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 76 points:

(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,28)(16,35)(17,40)(18,43)(19,44)(20,27)(21,30)(22,29)(23,37)(24,42)(25,36)(26,41)(31,65)(32,68)(33,69)(34,46)(38,72)(39,45)(47,74)(48,75)(49,76)(50,64)(51,71)(52,60)(53,58)(54,62)(55,67)(56,61)(57,66)(59,70)(63,73);(2,4)(6,9)(10,11)(13,15,29,58,18,32,38,63,41,66,71,75,42,67,16,30,20,34,37,62,17,31,59,76,19,33,60,74,36,61,39,64)(14,21,45,55,25,48,52,57,44,73,70,68,40,53,23,28,27,50,35,56,24,47,51,69,26,49,72,65,43,54,22,46);(5,7,8,12)(6,10,9,11)(13,16,36,71,19,38,17,29,20,39,42,60,41,59,18,37)(14,22,43,72,26,51,24,35,27,23,40,70,44,52,25,45)(15,30,61,75,33,63,31,58,34,64,67,74,66,76,32,62)(21,46,54,65,49,69,47,56,50,28,53,68,73,57,48,55)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=ehi,d^2=gh,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,h^a=hi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60212);