#490486

order	512
nilpotency class	4
isoclinism	isoclinic class of #490485
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1667
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

497 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,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^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, 397682);