#501149

order	512
nilpotency class	4
isoclinism	isoclinic class of #501144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24404
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1044 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,g^c=gi,e^d=ei,f^d=fhi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408345);