#145058

order	512
nilpotency class	4
isoclinism	isoclinic class of #145058
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10412
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

278 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,13,17,8)(4,19)(5,24)(6,30)(7,10)(9,21)(11,28)(12,27)(14,45,47,62)(15,33,18,35)(16,49,51,40)(20,36,55,53)(22,57)(23,38)(25,44,48,34)(26,54)(29,59)(31,42)(32,46,43,52)(37,60)(39,61)(41,56)(50,64,58,63);(1,3)(2,8)(4,20)(5,25)(6,15)(7,17)(9,36)(10,13)(11,40)(12,33)(14,22)(16,28)(18,27)(19,53)(21,55)(23,58)(24,34)(26,52)(29,48)(30,35)(31,50)(32,37)(38,64)(39,49)(41,45)(42,63)(43,54)(44,59)(46,60)(47,56)(51,61)(57,62);(1,4)(2,9,10,19)(3,14)(5,26)(6,31)(7,21)(8,32,13,43)(11,41,39,57)(12,42,30,38)(15,46)(16,50)(17,47)(18,52)(20,25)(22,28)(23,27)(24,37,59,54)(29,60)(33,62,35,45)(34,63,44,64)(36,49,53,40)(48,55)(51,58)(56,61)

Polycyclic presentation:

2^9|a^2=g,c^2=d,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=egi,f^c=fh,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, 52254);