Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382477 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2310 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14342 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,9)(4,19,22,37)(5,11)(6,12)(7,29)(8,13)(10,23,36,28)(14,41,43,34)(15,31)(16,32)(17,49)(18,33)(20,51,48,30)(21,50,24,54)(25,39)(26,57)(27,40)(35,53,38,56)(42,63,44,62)(45,60)(46,64)(47,61)(52,59,55,58);(1,3,6,16,8,18,7,17)(2,9,12,32,13,33,29,49)(4,20,23,43,22,48,28,14)(5,15,25,45,27,47,26,46)(10,34,19,30,36,41,37,51)(11,31,39,60,40,61,57,64)(21,52,53,44,24,55,56,42)(35,62,50,58,38,63,54,59);(1,4)(2,10)(3,14,18,43)(5,24,27,21)(6,28)(7,23)(8,22)(9,30,33,51)(11,38,40,35)(12,37)(13,36)(15,44)(16,48,17,20)(19,29)(25,53,26,56)(31,59)(32,34,49,41)(39,50,57,54)(42,47)(45,52)(46,55)(58,61)(60,63)(62,64);(1,5)(2,11)(3,15)(4,21)(6,25)(7,26)(8,27)(9,31)(10,35)(12,39)(13,40)(14,42)(16,45)(17,46)(18,47)(19,50)(20,52)(22,24)(23,53)(28,56)(29,57)(30,58)(32,60)(33,61)(34,62)(36,38)(37,54)(41,63)(43,44)(48,55)(49,64)(51,59)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289932);