#146054

order	512
nilpotency class	4
isoclinism	isoclinic class of #146054
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10621
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

214 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=d,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fh,h^b=hi,e^c=eg,f^c=fh,g^c=gi,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, 53250);