#512163

order	512
nilpotency class	4
isoclinism	isoclinic class of #511842
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#33021
prime radical	#23995	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

769 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)(3,13)(4,18)(5,22)(6,11)(7,12)(8,16)(9,21)(10,24)(14,38)(15,36)(17,37)(19,33)(20,50)(23,47)(25,30)(26,34)(27,55)(28,42)(29,44)(31,54)(32,52)(35,40)(39,46)(41,61)(43,60)(45,59)(48,49)(51,53)(56,64)(57,63)(58,62);(1,3)(2,8,12,28)(4,14)(5,16)(6,17)(7,15)(9,27,30,56)(10,13,22,36)(11,29,34,57)(18,46,33,58)(19,41)(20,39)(21,40)(23,44)(24,42)(25,45)(26,43)(31,35,53,59)(32,37,47,60)(38,50,61,48)(49,62)(51,55)(52,63)(54,64);(1,4,5,20,7,19,24,49)(2,9,10,31,12,30,22,53)(3,14,16,39,15,41,42,62)(6,25,23,54,26,21,52,51)(8,27,13,35,28,56,36,59)(11,33,32,50,34,18,47,48)(17,45,44,64,43,40,63,55)(29,58,37,61,57,46,60,38);(3,15)(4,19)(8,28)(9,30)(11,34)(13,36)(16,42)(17,43)(20,49)(21,25)(31,53)(32,47)(38,61)(44,63)(46,58)(51,54)

Polycyclic presentation:

2^9|b^2=h,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,d^b=di,d^c=di,f^c=fi,g^c=ghi,f^d=fh,g^f=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, 419359);