#153345

order	512
nilpotency class	5
isoclinism	isoclinic class of #153105
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

208 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,c^2=i,d^2=gh,e^2=g,f^2=gi,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,f^b=fi,d^c=dg,e^c=egh,f^c=fghi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60541);