#460462

order	512
nilpotency class	3
isoclinism	isoclinic class of #460462
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2499
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16364
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=e,f^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,g^b=gi,d^c=de,g^c=gi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367658);