#434710

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15506
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1049 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)(3,13,16,8)(4,18)(5,23,24,39)(6,12)(7,11)(9,21)(10,25,36,27)(14,32,19,29)(15,42)(17,38,26,31)(20,50,51,56)(22,47)(28,37)(30,44)(33,54,46,52)(34,53)(35,45)(40,61)(41,60,49,58)(43,55)(48,64)(57,62)(59,63);(1,3)(2,8)(4,19)(5,15)(6,26)(7,16)(9,32)(10,35)(11,13)(12,38)(14,21)(17,28)(18,29)(20,40)(22,41)(23,55)(24,43)(25,45)(27,44)(30,36)(31,37)(33,59)(34,58)(39,42)(46,57)(47,60)(48,51)(49,53)(50,61)(52,63)(54,62)(56,64);(1,4)(2,9,11,18)(3,14,16,19)(5,20)(6,22)(7,21)(8,29)(10,33,36,46)(12,34,37,47)(13,32)(15,40,43,48)(17,41,26,49)(23,56,39,50)(24,51)(25,52)(27,54)(28,53)(30,57)(31,58)(35,59)(38,60)(42,61)(44,62,45,63)(55,64);(1,5)(2,10)(3,15)(4,20)(6,27)(7,24)(8,30)(9,33)(11,36)(12,39)(13,35)(14,40)(16,43)(17,45)(18,46)(19,48)(21,51)(22,54)(23,37)(25,28)(26,44)(29,57)(31,42)(32,59)(34,50)(38,55)(41,63)(47,56)(49,62)(52,53)(58,61)(60,64)

Polycyclic presentation:

2^9|a^2=fg,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,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, 341906);