#499029

order	512
nilpotency class	4
isoclinism	isoclinic class of #499029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#482
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2033
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1039 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)(4,19)(5,25)(6,28)(7,12)(8,15)(9,23)(10,27)(11,26)(14,44)(16,33)(17,34)(18,35)(20,54)(21,52)(22,53)(24,40)(29,39)(30,38)(31,58)(32,60)(36,56)(37,57)(41,59)(42,46)(43,49)(45,62)(47,63)(48,61)(50,51)(55,64);(1,3)(2,8)(4,20)(5,17)(6,29)(7,18)(9,32)(10,35)(11,39)(12,34)(13,25)(14,21)(15,27)(16,31)(19,51)(22,45)(23,46)(24,50)(26,47)(28,43)(30,48)(33,41)(36,61)(37,64)(38,42)(40,44)(49,59)(52,62)(53,54)(55,56)(57,60)(58,63);(1,4,5,22,7,21,27,24)(2,9,10,37,12,36,25,38)(3,14,15,45,18,20,17,50)(6,30,26,56,31,57,59,23)(8,32,13,42,34,61,35,64)(11,40,28,52,41,53,58,19)(16,48,47,46,29,60,49,55)(33,54,43,62,39,44,63,51);(4,21)(9,36)(14,20)(19,52)(22,24)(23,56)(30,57)(32,61)(37,38)(40,53)(42,64)(44,54)(45,50)(46,55)(48,60)(51,62)

Polycyclic presentation:

2^9|c^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=di,f^c=fhi,g^c=gi,h^c=hi

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, 406225);