#509785

order	512
nilpotency class	4
isoclinism	isoclinic class of #509783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31000
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33540	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416981);