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 | |
3641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(3,9)(7,14)(10,12)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,39,42)(35,43,46,36)(37,40,49,53)(38,51,52,41)(44,55,57,47)(45,48,59,56)(50,62,61,54)(58,60,64,63);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(18,22)(23,25)(24,29)(30,31)(37,49)(40,53)(44,57)(47,55)(50,61)(54,62)(58,64)(60,63);(2,6)(7,14)(8,11)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,39,46)(34,36,42,43)(37,44,49,57)(38,45,52,59)(40,47,53,55)(41,48,51,56)(50,58,61,64)(54,60,62,63);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36)(34,35)(37,47)(38,48)(39,43)(40,44)(41,45)(42,46)(49,55)(50,60)(51,59)(52,56)(53,57)(54,58)(61,63)(62,64);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,40)(35,44)(36,47)(38,50)(39,49)(41,54)(42,53)(43,55)(45,58)(46,57)(48,60)(51,62)(52,61)(56,63)(59,64);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(33,38,39,52)(34,41,42,51)(35,45,46,59)(36,48,43,56)(37,50,49,61)(40,54,53,62)(44,58,57,64)(47,60,55,63)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=gi,e^2=g,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^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388579);