#417273

order	512
nilpotency class	3
isoclinism	isoclinic class of #394328
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2617
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15245
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

948 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)(3,14)(4,20)(5,26)(6,30)(7,12)(8,13)(9,17)(10,24)(11,29)(15,31)(16,46)(18,33)(19,42)(21,34)(22,51)(23,37)(25,36)(27,38)(28,39)(32,48)(35,55)(40,44)(41,49)(43,58)(45,50)(47,56)(52,59)(53,57)(54,60)(61,63)(62,64);(1,3,7,18,8,19,6,17)(2,9,30,42,13,33,12,14)(4,21,23,45,25,44,24,15)(5,16,28,48,29,49,27,47)(10,34,36,31,37,40,20,50)(11,32,39,46,26,56,38,41)(22,52,53,62,54,63,55,43)(35,59,60,58,57,61,51,64);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31)(11,35)(12,36)(13,37)(14,40)(16,43)(17,21)(18,44)(19,45)(20,30)(26,57)(27,53)(28,55)(29,54)(32,58)(33,50)(34,42)(38,60)(39,51)(41,61)(46,59)(47,52)(48,63)(49,62)(56,64);(1,5,6,27,8,29,7,28)(2,11,30,39,13,26,12,38)(3,16,17,47,19,49,18,48)(4,22,23,53,25,54,24,55)(9,32,42,46,33,56,14,41)(10,35,20,51,37,57,36,60)(15,43,21,52,45,62,44,63)(31,58,34,59,50,64,40,61)

Polycyclic presentation:

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

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, 324469);