#145440

order	512
nilpotency class	4
isoclinism	isoclinic class of #145440
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10480
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52636);