#145447

order	512
nilpotency class	4
isoclinism	isoclinic class of #145447
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10481
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

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

normal subgroups only

This group has 251 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:

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

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=fg,g^a=gi,c^b=cf,d^b=dgi,e^b=egi,f^b=fgi,g^b=gi,h^b=hi,d^c=di,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, 52643);