#488934

order	512
nilpotency class	4
isoclinism	isoclinic class of #488933
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1629
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3293	C₂
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

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

normal subgroups only

This group has 889 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)(4,17,20,52)(5,22)(6,28)(7,10)(9,19,34,55)(11,38)(12,25)(13,43,46,61)(14,48)(15,42)(16,31)(18,40,21,53)(23,56)(24,36,54,35)(26,37)(27,41)(29,33)(30,45,58,64)(32,57)(39,50)(44,51,47,62)(49,60,63,59);(1,3,7,16)(2,8,10,31)(4,13,20,46)(5,23,27,57)(6,29,25,15)(9,30,34,58)(11,39,37,48)(12,42,28,33)(14,38,50,26)(17,51,52,62)(18,47,21,44)(19,49,55,63)(22,56,41,32)(24,45,54,64)(35,60,36,59)(40,43,53,61);(1,4)(2,9)(3,13)(5,24)(6,18)(7,20)(8,30)(10,34)(11,40)(12,35)(14,49)(15,44)(16,46)(17,41)(19,26)(21,25)(22,52)(23,45)(27,54)(28,36)(29,47)(31,58)(32,51)(33,59)(37,53)(38,55)(39,43)(42,60)(48,61)(50,63)(56,62)(57,64);(2,10)(8,31)(9,34)(11,37)(12,28)(17,52)(22,41)(30,58)(32,56)(33,42)(35,36)(39,48)(40,53)(43,61)(51,62)(59,60)

Polycyclic presentation:

2^9|a^2=g,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=fhi,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, 396130);