#400119

order	512
nilpotency class	3
isoclinism	isoclinic class of #393928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14796
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1407 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)(3,9)(4,20,23,37)(5,26,27,39)(6,12)(7,13)(8,14)(10,24,36,25)(11,29,38,28)(15,34,42,59)(16,44,45,55)(17,32)(18,40)(19,33)(21,30,43,48)(22,51)(31,46,58,47)(35,54)(41,57)(49,52)(50,64)(53,56)(60,62)(61,63);(1,3)(2,9)(4,21)(5,16)(6,17)(7,18)(8,19)(10,34)(11,31)(12,32)(13,40)(14,33)(15,24)(20,48)(22,50)(23,43)(25,42)(26,55)(27,45)(28,46)(29,47)(30,37)(35,60)(36,59)(38,58)(39,44)(41,53)(49,63)(51,64)(52,61)(54,62)(56,57);(1,4)(2,10)(3,15)(5,22)(6,25)(7,24)(8,23)(9,30)(11,35)(12,37)(13,20)(14,36)(16,41)(17,21)(18,43)(19,42)(26,56)(27,54)(28,53)(29,52)(31,57)(32,59)(33,48)(34,40)(38,51)(39,49)(44,60)(45,61)(46,62)(47,50)(55,64)(58,63);(1,5,6,29,8,27,7,28)(2,11,12,26,14,38,13,39)(3,16,17,47,19,45,18,46)(4,22,25,52,23,54,24,53)(9,31,32,55,33,58,40,44)(10,35,37,56,36,51,20,49)(15,41,21,50,42,61,43,62)(30,57,59,64,48,63,34,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307315);