#153402

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

Polycyclic presentation:

2^9|a^2=di,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, 60598);