#153389

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

186 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 78 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)(7,12)(8,10)(9,11)(13,14,17,23,31,35,42,36,19,20,32,34,29,40,44,37,16,22,28,39,27,41,43,33,15,25,26,38,18,24,30,21);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,20,22,25)(17,26,28,32)(18,27,29,31)(21,36,37,33)(23,34,39,38)(24,35,40,41)(30,43,44,42);(2,4)(5,8,9,7)(6,10,11,12)(14,21)(15,19)(17,30)(18,31)(20,33)(22,37)(23,24)(25,36)(26,42)(27,29)(28,44)(32,43)(34,41)(35,38)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=d,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, 60585);