#153311

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, 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

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

normal subgroups only

This group has 266 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) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,4)(9,10,18,26)(11,27,33,62)(12,34,30,57)(13,22,35,23)(14,21,15,43)(16,46,40,72)(17,49,48,25)(19,44,55,29)(20,36,58,31)(24,64,59,28)(32,69,50,66)(37,54,67,65)(38,53,68,70)(39,63,41,61)(42,60,45,56)(47,51,71,52);(5,6)(7,8)(9,11,18,33)(10,19,26,55)(12,35,30,13)(14,29,15,44)(16,47,40,71)(17,50,48,32)(20,56,58,60)(21,51,43,52)(22,27,23,62)(24,34,59,57)(25,65,49,54)(28,31,64,36)(37,45,67,42)(38,39,68,41)(46,53,72,70)(61,69,63,66);(2,4)(6,8)(9,12,13,11,28,29,42,47,17,38,39,32,40,67,15,36,18,30,35,33,64,44,45,71,48,68,41,50,16,37,14,31)(10,20,21,54,46,66,61,70,25,52,56,55,59,62,23,57,26,58,43,65,72,69,63,53,49,51,60,19,24,27,22,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,d^2=gh,e^2=g,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fgh,g^b=gh,h^b=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, 60507);