#499121

order	512
nilpotency class	4
isoclinism	isoclinic class of #499120
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2044
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1067 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)(5,23)(6,12)(7,10)(8,15)(9,21)(11,26)(14,43)(16,42)(17,37)(19,51)(20,53)(22,40)(24,31)(25,49)(27,60)(28,34)(29,36)(30,61)(32,46)(33,63)(35,59)(38,56)(39,58)(41,45)(44,62)(47,64)(48,52)(50,55)(54,57);(1,3,7,17)(2,8,10,31)(4,19,22,14)(5,24,26,15)(6,27,29,16)(9,33,34,30)(11,37,23,13)(12,39,36,32)(18,48,40,62)(20,44,55,52)(21,47,28,45)(25,46,56,58)(35,41,57,64)(38,42,49,60)(43,53,51,50)(54,61,59,63);(1,4,5,20,7,22,26,55)(2,9,11,35,10,34,23,57)(3,14,15,44,17,19,24,52)(6,28,25,59,29,21,56,54)(8,30,13,41,31,33,37,64)(12,40,38,53,36,18,49,50)(16,47,46,63,27,45,58,61)(32,48,42,51,39,62,60,43);(2,10)(8,31)(9,34)(11,23)(12,36)(13,37)(18,40)(30,33)(32,39)(35,57)(38,49)(41,64)(42,60)(43,51)(48,62)(50,53)

Polycyclic presentation:

2^9|b^2=i,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,f^c=fi,g^c=gi,h^c=hi,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, 406317);