#153401

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×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

231 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6,8,10)(7,11,12,9)(13,14,16,21,29,37,43,41,20,24,32,39,35,25,33,28,18,15,17,22,30,38,44,42,36,26,34,40,19,23,31,27);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,27)(17,28)(19,37)(20,26)(21,31)(22,33)(23,29)(24,36)(25,30)(32,42)(34,41)(35,38)(39,44)(40,43);(2,4)(13,15,18,14)(16,28,17,27)(19,38,35,37)(20,24,36,26)(21,31,22,33)(23,29,25,30)(32,41,34,42)(39,44,40,43)

Polycyclic presentation:

2^9|a^2=d,b^2=degi,c^2=i,d^2=gh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dg,e^c=eg,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60597);