#145559

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	#10539
prime radical	#3147	C₂²
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

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

normal subgroups only

This group has 266 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=d,d^2=gi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gi,e^c=eghi,f^c=fgh,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, 52755);