#151643

order	512
nilpotency class	5
isoclinism	isoclinic class of #151643
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12430
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2170 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

149 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 149 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=df,d^2=fh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,h^a=hi,e^c=eg,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225561, #72225565, #72225569
maximal normal subgroup of	#72225561, #72225565, #72225569

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58839);