#507330

order	512
nilpotency class	4
isoclinism	isoclinic class of #507330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29051
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3060	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

441 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)(4,9)(5,28)(6,13)(7,14)(8,19)(10,16)(11,30)(12,31)(17,50)(18,43)(20,36)(21,42)(22,56)(23,59)(24,45)(25,38)(26,39)(27,41)(29,47)(32,49)(33,53)(34,48)(35,54)(37,57)(40,60)(44,61)(46,52)(51,62)(55,63)(58,64);(1,3,16,48,6,19,31,21,7,20,49,18,30,54,5,29)(2,8,32,47,13,36,28,34,14,35,10,42,11,15,12,43)(4,22,23,52,26,55,60,51,27,58,61,17,24,57,25,53)(9,37,44,62,39,56,38,50,41,63,59,33,45,64,40,46);(1,4)(2,9)(3,17)(5,23)(6,24)(7,27)(8,33)(10,38)(11,39)(12,44)(13,45)(14,41)(15,46)(16,25)(18,22)(19,51)(20,52)(21,55)(26,30)(28,59)(29,57)(31,61)(32,40)(34,56)(35,62)(36,50)(37,42)(43,64)(47,63)(48,58)(49,60)(53,54);(1,5,6,16,7,31,30,49)(2,10,11,28,14,32,13,12)(3,18,54,21,20,48,19,29)(4,23,24,25,27,61,26,60)(8,34,36,42,35,43,15,47)(9,38,39,59,41,40,45,44)(17,22,53,55,52,58,51,57)(33,56,50,37,62,64,46,63)

Polycyclic presentation:

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

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, 414526);