#451621

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16005
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358817);