#153398

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

Polycyclic presentation:

2^9|a^2=di,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, 60594);