#510152

order	512
nilpotency class	4
isoclinism	isoclinic class of #510151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31322
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#444 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

440 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)(3,15)(4,22)(5,10)(6,29)(7,12)(8,14)(9,18)(11,24)(13,31)(16,43)(17,47)(19,34)(20,35)(21,41)(23,52)(25,56)(26,39)(27,40)(28,42)(30,36)(32,46)(33,48)(37,50)(38,57)(44,61)(45,59)(49,64)(51,58)(53,62)(54,63)(55,60);(1,3)(2,9,14,15)(4,23,26,17)(5,21)(6,16)(7,19)(8,18)(10,34,12,41)(11,37)(13,43,29,59)(20,30)(22,47)(24,48,27,50)(25,53,58,44)(28,46)(31,45)(32,42,35,36)(33,40)(38,63)(39,52)(49,55,54,57)(51,62)(56,61)(60,64);(2,10)(3,16,18,45)(5,7)(6,28)(9,32,15,35)(11,22)(12,14)(13,29)(17,44,23,53)(19,46,21,20)(24,27)(25,57)(30,31)(33,61,37,62)(34,43,41,59)(38,60)(39,40)(47,63,52,64)(48,49,50,54)(55,58);(1,4,8,26)(2,11,14,40)(3,17,18,23)(5,27,7,24)(6,25,31,58)(9,33,15,37)(10,22,12,39)(13,38,29,60)(16,44,45,53)(19,50,21,48)(20,49,46,54)(28,57,30,55)(32,61,35,62)(34,52,41,47)(36,51,42,56)(43,64,59,63)

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417348);