#145445

order	512
nilpotency class	4
isoclinism	isoclinic class of #144056
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1697
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10481
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

254 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=dh,d^2=g,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dgi,e^b=eg,f^b=fgi,g^b=gi,h^b=hi,d^c=di,e^c=eghi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52641);