#153400

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

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

normal subgroups only

This group has 223 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)(13,14,16,22,33,47,63,41,18,24,34,48,38,52,57,42,19,25,35,49,60,72,67,53,37,51,62,40,17,23,31,20)(15,26,27,50,36,69,65,44,29,55,58,73,61,74,66,45,30,56,59,76,70,75,68,54,39,71,64,43,28,46,32,21);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,20)(16,31)(17,33)(18,37)(21,26)(22,23)(24,53)(25,42)(27,32)(28,36)(29,39)(34,67)(35,57)(38,60)(40,47)(41,51)(43,69)(44,71)(45,56)(46,50)(48,72)(49,52)(54,55)(58,68)(59,66)(61,70)(62,63)(64,65)(73,75)(74,76);(2,4)(13,15,19,30)(14,21,25,45)(16,32,35,66)(17,36,38,70)(18,39,37,29)(20,26,42,56)(22,46,49,74)(23,50,52,76)(24,54,51,44)(27,57,59,31)(28,60,61,33)(34,68,62,65)(40,69,48,75)(41,71,53,55)(43,72,73,47)(58,63,64,67)

Polycyclic presentation:

2^9|a^2=d,b^2=deg,c^2=i,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, 60596);