#511850

order	512
nilpotency class	4
isoclinism	isoclinic class of #511849
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32899
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2026 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)(3,14)(4,9)(5,11)(6,10)(7,13)(8,16)(12,25)(15,42)(17,40)(18,39)(19,31)(20,30)(21,32)(22,36)(23,33)(24,41)(26,44)(27,53)(28,46)(29,49)(34,48)(35,52)(37,47)(38,43)(45,59)(50,58)(51,60)(54,62)(55,64)(56,61)(57,63);(1,3,7,17)(2,8,13,28)(4,15,19,43)(5,16,22,46)(6,24,25,18)(9,26,31,56)(10,27,12,37)(11,14,36,40)(20,44,48,61)(21,51,49,45)(23,53,52,47)(29,54,32,55)(30,42,34,38)(33,41,35,39)(50,64,63,62)(57,59,58,60);(1,4)(2,9)(3,15)(5,20)(6,21)(7,19)(8,26)(10,32)(11,34)(12,29)(13,31)(14,38)(16,44)(17,43)(18,45)(22,48)(23,50)(24,51)(25,49)(27,55)(28,56)(30,36)(33,57)(35,58)(37,54)(39,60)(40,42)(41,59)(46,61)(47,62)(52,63)(53,64);(2,10,13,12)(4,19)(5,22)(8,27,28,37)(9,29,31,32)(11,35,36,33)(14,39,40,41)(15,43)(16,46)(21,49)(23,52)(26,54,56,55)(30,58,34,57)(38,59,42,60)(45,51)(47,53)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419046);