Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405381 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14912 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,40)(4,20,24,10)(5,11,27,38)(6,13,7,12)(9,29,32,18)(16,33,43,46)(17,41,45,60)(21,30,49,55)(22,34,52,47)(23,35,25,36)(26,39,28,37)(31,53,56,44)(42,61,62,57)(48,63,64,54)(50,59,51,58);(1,3)(2,9)(4,21,24,49)(5,17)(6,29)(7,18)(8,19)(10,33,20,46)(11,31)(12,40)(13,15)(14,32)(16,25,43,23)(22,48,52,64)(26,53)(27,45)(28,44)(30,36,55,35)(34,57,47,61)(37,60)(38,56)(39,41)(42,51,62,50)(54,59,63,58);(1,4,6,23,8,24,7,25)(2,10,12,35,14,20,13,36)(3,16,18,21,19,43,29,49)(5,22,26,50,27,52,28,51)(9,30,15,33,32,55,40,46)(11,34,37,58,38,47,39,59)(17,42,44,48,45,62,53,64)(31,54,41,57,56,63,60,61);(1,5)(2,11)(3,17)(4,22)(6,26)(7,28)(8,27)(9,31)(10,34)(12,37)(13,39)(14,38)(15,41)(16,42)(18,44)(19,45)(20,47)(21,48)(23,50)(24,52)(25,51)(29,53)(30,54)(32,56)(33,57)(35,58)(36,59)(40,60)(43,62)(46,61)(49,64)(55,63)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312578);