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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3707 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)(3,4)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(18,21)(23,26)(24,29)(30,32)(33,34,38,42)(35,39,45,54)(36,40,49,47)(37,41,51,58)(43,52,46,55)(44,53,50,57)(48,56,62,61)(59,63,60,64);(2,7)(4,8)(9,13)(12,15)(20,28)(24,29)(25,31)(30,32)(33,35,38,45)(34,39,42,54)(36,46,49,43)(37,44,51,50)(40,55,47,52)(41,53,58,57)(48,60,62,59)(56,64,61,63);(1,2,6,7)(3,8,11,4)(5,9,14,13)(10,15,16,12)(17,18,22,21)(19,23,27,26)(20,24,28,29)(25,30,31,32)(33,35)(34,39)(36,43)(37,50)(38,45)(40,52)(41,57)(42,54)(44,51)(46,49)(47,55)(48,60)(53,58)(56,64)(59,62)(61,63);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,13)(14,16)(17,19,22,27)(18,23,21,26)(20,25,28,31)(24,30,29,32)(33,34)(35,39)(36,47)(37,41)(38,42)(40,49)(43,55)(44,53)(45,54)(46,52)(48,61)(50,57)(51,58)(56,62)(59,64)(60,63);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,20)(18,24)(19,25)(21,29)(22,28)(23,30)(26,32)(27,31)(33,36)(34,40)(35,43)(37,48)(38,49)(39,52)(41,56)(42,47)(44,59)(45,46)(50,60)(51,62)(53,63)(54,55)(57,64)(58,61);(1,5,6,14)(2,9,7,13)(3,10,11,16)(4,12,8,15)(17,21,22,18)(19,26,27,23)(20,29,28,24)(25,32,31,30)(33,37)(34,41)(35,44)(36,48)(38,51)(39,53)(40,56)(42,58)(43,59)(45,50)(46,60)(47,61)(49,62)(52,63)(54,57)(55,64)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833855);