#153397

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	C₂²×D₁₆	C₂²
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

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

normal subgroups only

This group has 249 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=deg,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, 60593);