Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2352 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13881 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1173 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)(3,14)(4,20,25,37)(5,26)(6,30)(7,12)(8,13)(9,17)(10,24,38,23)(11,27)(15,41,44,31)(16,45)(18,34)(19,33)(21,50,43,35)(22,52,55,36)(28,39)(29,56)(32,48)(40,46)(42,62,64,59)(47,58)(49,54,60,53)(51,57,63,61);(1,3,6,17,8,19,7,18)(2,9,12,14,13,34,30,33)(4,21,23,15,25,43,24,44)(5,16,27,46,29,48,28,47)(10,35,20,41,38,50,37,31)(11,32,26,40,39,45,56,58)(22,51,53,42,55,63,54,64)(36,59,49,61,52,62,60,57);(1,4)(2,10,13,38)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31,34,41)(11,36,39,52)(12,20,30,37)(14,35,33,50)(16,42)(17,43)(18,21)(19,44)(26,49,56,60)(27,53)(28,54)(29,55)(32,57,45,61)(40,59,58,62)(46,63)(47,51)(48,64);(1,5,6,27,8,29,7,28)(2,11,12,26,13,39,30,56)(3,16,17,46,19,48,18,47)(4,22,23,53,25,55,24,54)(9,32,14,40,34,45,33,58)(10,36,20,49,38,52,37,60)(15,42,43,63,44,64,21,51)(31,57,35,59,41,61,50,62)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268023);