Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| 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 | |
3713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,20)(19,26,22,24)(21,28,30,25)(27,31,29,32)(33,34,38,39)(35,43,46,40)(36,41,49,47)(37,42,51,52)(44,57,59,53)(45,56,50,54)(48,55,62,60)(58,64,61,63);(2,7)(3,10)(9,12)(11,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(34,39)(40,43)(41,47)(42,52)(53,57)(54,56)(55,60)(63,64);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,35)(34,40)(36,44)(37,50)(38,46)(39,43)(41,53)(42,56)(45,51)(47,57)(48,61)(49,59)(52,54)(55,64)(58,62)(60,63);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,36,38,49)(34,41,39,47)(35,44,46,59)(37,48,51,62)(40,53,43,57)(42,55,52,60)(45,58,50,61)(54,63,56,64);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,19,23,22)(18,26,20,24)(21,27,30,29)(25,32,28,31)(33,34)(35,40)(36,47)(37,42)(38,39)(41,49)(43,46)(44,57)(45,54)(48,60)(50,56)(51,52)(53,59)(55,62)(58,64)(61,63);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,37,38,51)(34,42,39,52)(35,45,46,50)(36,48,49,62)(40,54,43,56)(41,55,47,60)(44,58,59,61)(53,63,57,64)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324235);