#511462

order	512
nilpotency class	4
isoclinism	isoclinic class of #511461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32584
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

370 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:

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

Polycyclic presentation:

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

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, 418658);