#460311

order	512
nilpotency class	3
isoclinism	isoclinic class of #460311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16361
prime radical	#60422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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)(4,20,23,37,24,38,29,10)(5,25,27,40,28,41,26,11)(15,35,21,42,46,58,45,31)(16,47,48,43,49,56,50,32)(22,36,54,51,55,39,53,61)(44,59,52,62,64,57,63,60);(1,3)(2,9)(4,21)(5,16)(6,17)(7,18)(8,19)(10,35)(11,32)(12,14)(13,33)(15,23)(20,31)(22,52)(24,45)(25,56)(26,48)(27,50)(28,49)(29,46)(30,34)(36,60)(37,58)(38,42)(39,62)(40,43)(41,47)(44,54)(51,57)(53,64)(55,63)(59,61);(1,4)(2,10)(3,15)(5,22)(6,29)(7,23)(8,24)(9,31)(11,39)(12,37)(13,38)(14,42)(16,44)(17,21)(18,45)(19,46)(20,30)(25,51)(26,54)(27,53)(28,55)(32,59)(33,35)(34,58)(36,40)(41,61)(43,57)(47,60)(48,63)(49,64)(50,52)(56,62);(1,5)(2,11,12,40)(3,16)(4,22)(6,27)(7,26)(8,28)(9,32,14,43)(10,36,37,39)(13,25,30,41)(15,44)(17,50)(18,48)(19,49)(20,51,38,61)(21,52)(23,54)(24,55)(29,53)(31,57,42,59)(33,56,34,47)(35,60,58,62)(45,63)(46,64)

Polycyclic presentation:

2^9|a^2=g,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,f^c=fi,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, 367507);