Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60942 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,5,10)(3,13,15,23,17,26,16,8)(4,18,20,9,21,24,19,25)(14,22,28,27,30,32,29,31)(33,34)(35,45,49,55)(36,50,53,57)(37,43)(38,42)(39,44)(40,47,56,48)(41,51,58,52)(46,60)(54,63)(59,62)(61,64);(1,3)(2,8,12,23)(4,14)(5,16)(6,15)(7,17)(9,22,25,32)(10,26,11,13)(18,31,24,27)(19,29)(20,28)(21,30)(33,35,37,48,39,49,38,47)(34,40,43,55,44,56,42,45)(36,46,51,62,53,63,52,61)(41,54,50,64,58,60,57,59);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,18)(11,24)(12,25)(13,27)(15,28)(16,29)(17,30)(23,32)(26,31)(33,36)(34,41)(35,46)(37,51)(38,52)(39,53)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,62)(49,63)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,23,26)(9,18,25,24)(14,28,30,29)(22,27,32,31)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,53,52)(40,55,56,45)(41,57,58,50)(46,61,63,62)(54,64,60,59)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105783);