#419856

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15273
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1541 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 327052);