#10247021

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3643 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)(3,5,10,8)(4,9,14,16)(11,15,12,13)(34,39)(37,50)(40,44)(41,52)(46,59)(49,51)(54,56)(61,62);(2,7)(3,10)(9,16)(11,12)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(34,39)(35,43)(36,47)(42,53)(46,59)(49,51)(54,56)(63,64);(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,34,38,39)(35,44,43,40)(36,41,47,52)(37,42,50,53)(45,56,48,54)(46,60,59,55)(49,57,51,58)(61,64,62,63);(1,3,6,10)(2,8,7,5)(4,12,14,11)(9,15,16,13)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,35,38,43)(34,40,39,44)(36,48,47,45)(37,46,50,59)(41,56,52,54)(42,55,53,60)(49,62,51,61)(57,64,58,63);(1,4,6,14)(2,9,7,16)(3,11,10,12)(5,15,8,13)(17,19)(18,20)(21,29)(22,27)(23,25)(24,31)(26,30)(28,32)(33,36,38,47)(34,41,39,52)(35,45,43,48)(37,51,50,49)(40,54,44,56)(42,58,53,57)(46,62,59,61)(55,64,60,63);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,11,16,12)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37)(34,42)(35,46)(36,49)(38,50)(39,53)(40,55)(41,57)(43,59)(44,60)(45,61)(47,51)(48,62)(52,58)(54,63)(56,64)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=gi,e^2=gh,f^2=g,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, 10154217);