#9912907

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3682 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820103);