#413939

order	512
nilpotency class	3
isoclinism	isoclinic class of #413938
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15159
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1345 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)(3,14)(4,9)(5,23)(6,12)(7,13)(8,17)(10,26)(11,29)(15,42)(16,45)(18,33)(19,30)(20,51)(21,37)(22,38)(24,40)(25,39)(27,32)(28,36)(31,48)(34,44)(35,54)(41,47)(43,62)(46,55)(49,58)(50,61)(52,60)(53,56)(57,64)(59,63);(1,3)(2,8)(4,19)(5,16)(6,27)(7,18)(9,34)(10,31)(11,33)(12,14)(13,32)(15,21)(17,29)(20,50)(22,49)(23,55)(24,47)(25,48)(26,46)(28,44)(30,37)(35,59)(36,58)(38,42)(39,41)(40,45)(43,54)(51,62)(52,64)(53,63)(56,57)(60,61);(1,4,7,22)(2,9,13,38)(3,15,18,44)(5,20,25,53)(6,28,29,21)(8,30,32,58)(10,35,40,60)(11,37,12,36)(14,34,33,42)(16,43,48,64)(17,49,27,19)(23,56,39,51)(24,54,26,52)(31,57,45,62)(41,59,55,61)(46,50,47,63);(1,5,7,25)(2,10)(3,16,18,48)(4,20,22,53)(6,26,29,24)(8,31)(9,35)(11,23)(12,39)(13,40)(14,41)(15,43,44,64)(17,47,27,46)(19,50,49,63)(21,54,28,52)(30,57)(32,45)(33,55)(34,59)(36,51)(37,56)(38,60)(42,61)(58,62)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321135);