#435793

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15538
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

979 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=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, 342989);