#502098

order	512
nilpotency class	4
isoclinism	isoclinic class of #502097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25076
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

747 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409294);