#510364

order	512
nilpotency class	4
isoclinism	isoclinic class of #510363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31547
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1773 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

416 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417560);