#509296

order	512
nilpotency class	4
isoclinism	isoclinic class of #509294
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30567
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24809	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416492);