#490560

order	512
nilpotency class	4
isoclinism	isoclinic class of #490559
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1677
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
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

422 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fh,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, 397756);