#153367

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂⁴	D₁₆
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=di,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,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, 60563);