#153210

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

253 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 76
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 76 points:

(5,6,8,10)(7,11,12,9)(13,14,16,20)(15,25,26,54)(17,33,30,55)(18,38,31,63)(19,40,32,24)(21,45,42,27)(22,48,43,36)(23,51,44,35)(28,59,57,76)(29,60,41,56)(34,52,61,74)(37,70,62,66)(39,49,64,73)(46,71,72,68)(47,58,53,67)(50,69,65,75);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,20)(21,42)(22,43)(23,44)(24,40)(25,54)(33,55)(38,63)(46,72)(47,53)(49,73)(50,65)(52,74)(56,60)(59,76)(66,70);(2,4)(13,15,16,26)(14,21,20,42)(17,34,30,61)(18,39,31,64)(19,41,32,29)(22,49,43,73)(23,52,44,74)(24,53,40,47)(25,55,54,33)(27,48,45,36)(28,51,57,35)(37,71,62,68)(38,46,63,72)(50,76,65,59)(56,66,60,70)(58,69,67,75)

Polycyclic presentation:

2^9|a^2=d,b^2=di,c^2=i,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,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, 60406);