#10247221

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154417);