Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7339802 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #48088 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,14,18,40)(4,9)(5,10)(6,12)(7,13)(8,17,11,26)(15,45,48,43)(16,44,52,35)(19,31)(20,33)(21,34)(22,30,51,37)(23,36)(24,38)(25,39)(27,41)(28,42)(29,47,32,53)(46,63,50,61)(49,59)(54,58,56,60)(55,62)(57,64);(1,3)(2,8)(4,15)(5,22)(6,26)(7,18)(9,29)(10,35)(11,13)(12,40)(14,42)(16,49)(17,28)(19,54)(20,53)(21,48)(23,56)(24,52)(25,51)(27,47)(30,59)(31,61)(32,34)(33,43)(36,63)(37,38)(39,44)(41,45)(46,57)(50,55)(58,64)(60,62);(1,4,7,21)(2,9,13,34)(3,15,18,48)(5,23,25,19)(6,27,28,20)(8,29,11,32)(10,36,39,31)(12,41,42,33)(14,43,40,45)(16,50,52,46)(17,53,26,47)(22,54,51,56)(24,55,49,57)(30,60,37,58)(35,61,44,63)(38,62,59,64);(1,5,6,24,7,25,28,49)(2,10,12,38,13,39,42,59)(3,16,26,22,18,52,17,51)(4,19,20,55,21,23,27,57)(8,30,40,35,11,37,14,44)(9,31,33,62,34,36,41,64)(15,46,53,54,48,50,47,56)(29,58,43,61,32,60,45,63);(1,3)(2,11)(4,15)(5,16)(6,26)(7,18)(8,13)(9,32)(10,37)(12,14)(17,28)(19,46)(20,53)(21,48)(22,24)(23,50)(25,52)(27,47)(29,34)(30,39)(31,60)(33,45)(35,59)(36,58)(38,44)(40,42)(41,43)(49,51)(54,55)(56,57)(61,64)(62,63)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,h^d=hi,f^e=fi,g^e=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7247001);