#151645

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
prime radical	#12436	C₂
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

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

normal subgroups only

This group has 144 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)(4,22)(5,6)(7,8)(11,26)(12,40)(13,33)(14,32)(15,35)(16,34)(17,36)(19,50)(20,53)(21,49)(23,24)(25,39)(27,30)(28,29)(31,41)(37,46)(42,63)(43,60)(44,59)(45,58)(47,57)(51,52)(54,61)(55,64)(56,62);(1,2,11,37,6,14,25,35,7,15,23,32,29,46,4,10,9,17,22,33,28,45,24,34,8,16,39,58,5,13,26,36)(3,12,38,60,18,41,51,55,30,47,54,57,53,64,19,31,21,43,48,40,49,62,50,59,20,42,61,63,27,44,52,56);(1,3)(2,12)(4,19)(5,27)(6,18)(7,30)(8,20)(9,21)(10,31)(11,38)(13,44)(14,41)(15,47)(16,42)(17,43)(22,48)(23,54)(24,50)(25,51)(26,52)(28,49)(29,53)(32,57)(33,40)(34,59)(35,55)(36,56)(37,60)(39,61)(45,62)(46,64)(58,63)

Polycyclic presentation:

2^9|b^2=dfi,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	#72225563, #72225565, #72225569
maximal normal subgroup of	#72225563, #72225565, #72225569

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58841);