#391183

order	512
nilpotency class	3
isoclinism	isoclinic class of #32700
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2760
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14620
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

591 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298379);