#488871

order	512
nilpotency class	4
isoclinism	isoclinic class of #488870
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1619
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

948 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396067);