Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360368 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2344 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13904 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1197 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)(3,15,20,9)(4,21,25,40)(5,26,29,11)(6,13)(7,14)(8,12)(10,24,37,30)(16,43)(17,32)(18,34,19,33)(22,31)(23,52)(27,39,28,38)(35,45)(36,53)(41,49)(42,48)(44,60,63,62)(46,58)(47,59)(50,54)(51,61,64,57)(55,56);(1,3,7,19,8,20,6,18)(2,9,14,34,12,15,13,33)(4,22,24,45,25,49,30,16)(5,17,28,47,29,48,27,46)(10,35,21,41,37,43,40,31)(11,32,39,59,26,42,38,58)(23,51,53,63,54,64,56,44)(36,60,50,61,55,62,52,57);(1,4)(2,10,12,37)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31,15,41)(11,36,26,55)(13,21,14,40)(17,44)(18,22)(19,49)(20,45)(27,53)(28,56)(29,54)(32,57,42,61)(33,35,34,43)(38,50,39,52)(46,51)(47,64)(48,63)(58,60,59,62);(1,5)(2,11)(3,17)(4,23)(6,27)(7,28)(8,29)(9,32)(10,36)(12,26)(13,38)(14,39)(15,42)(16,44)(18,46)(19,47)(20,48)(21,50)(22,51)(24,53)(25,54)(30,56)(31,57)(33,58)(34,59)(35,60)(37,55)(40,52)(41,61)(43,62)(45,63)(49,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269060);