#490561

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	#33594	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

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

normal subgroups only

This group has 425 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)(3,17)(4,25)(5,12)(6,14)(7,13)(8,15)(9,16)(10,21)(11,28)(18,44)(19,50)(20,45)(22,36)(23,37)(24,48)(26,58)(27,43)(29,41)(30,34)(31,38)(32,40)(33,46)(35,53)(39,55)(42,49)(47,52)(51,61)(54,63)(56,60)(57,62)(59,64);(1,3,18,22,5,23,6,21,9,24,7,31,8,20,30,55)(2,10,34,37,15,36,13,17,16,39,14,45,12,38,44,48)(4,19,49,57,26,53,28,56,29,54,32,59,33,51,27,52)(11,35,58,62,42,50,25,47,43,61,46,64,40,63,41,60);(1,4)(2,11)(3,19)(5,26)(6,28)(7,32)(8,33)(9,29)(10,35)(12,40)(13,25)(14,46)(15,42)(16,43)(17,47)(18,49)(20,51)(21,56)(22,57)(23,53)(24,54)(27,30)(31,59)(34,58)(36,50)(37,62)(38,63)(39,61)(41,44)(45,64)(48,60)(52,55);(1,5,9,8)(2,12,16,15)(3,20,24,23)(4,26,29,33)(6,7,30,18)(10,36,39,38)(11,40,43,42)(13,34,44,14)(17,45,48,37)(19,51,54,53)(21,22,55,31)(25,58,41,46)(27,49,28,32)(35,50,61,63)(47,64,60,62)(52,59,56,57)

Polycyclic presentation:

2^9|b^2=eh,d^2=i,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, 397757);