#471501

order	512
nilpotency class	3
isoclinism	isoclinic class of #471500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16798
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

895 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378697);