#493466

order	512
nilpotency class	4
isoclinism	isoclinic class of #493464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1782
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2139 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)(3,13,17,36)(4,18)(5,10)(6,25,27,11)(7,12)(8,16)(9,21)(14,39,42,37)(15,38,22,52)(19,45)(20,32,51,46)(23,54,55,34)(24,35)(26,30)(28,50)(29,43)(31,49)(33,53)(40,60,47,59)(41,44)(48,58,64,62)(56,63)(57,61);(1,3,7,17)(2,8,12,30)(4,14,21,42)(5,22,24,15)(6,26,27,16)(9,28,18,44)(10,33,35,29)(11,36,25,13)(19,47,49,40)(20,50,51,41)(23,43,55,53)(31,57,45,56)(32,39,46,37)(34,38,54,52)(48,61,64,63)(58,59,62,60);(1,4)(2,9,12,18)(3,14)(5,19)(6,20)(7,21)(8,28,30,44)(10,31,35,45)(11,32,25,46)(13,37,36,39)(15,40)(16,41)(17,42)(22,47)(23,48)(24,49)(26,50)(27,51)(29,56,33,57)(34,58,54,62)(38,59,52,60)(43,61)(53,63)(55,64);(1,5,7,24)(2,10,12,35)(3,15,17,22)(4,19,21,49)(6,23,27,55)(8,29,30,33)(9,31,18,45)(11,34,25,54)(13,38,36,52)(14,40,42,47)(16,43,26,53)(20,48,51,64)(28,56,44,57)(32,58,46,62)(37,59,39,60)(41,61,50,63)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,g^b=ghi,f^c=fi,g^e=gi

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, 400662);