#143684

order	512
nilpotency class	4
isoclinism	isoclinic class of #143666
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19710	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 357 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=gi,e^2=h,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fghi,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fghi,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, 50880);