#490555

order	512
nilpotency class	4
isoclinism	isoclinic class of #490555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1675
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

382 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=eg,e^2=gi,f^2=h,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^c=fhi,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, 397751);