Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,10,16,11,17,12,18)(7,13,21,36,22,37,33,44)(8,23,25,39,26,40,27,14)(9,28,31,15,30,43,32,42)(19,45,47,60,48,63,53,34)(20,49,52,62,51,58,54,35)(24,41,56,55,29,38,57,64)(46,59,66,68,50,61,67,65);(1,2,3,4)(5,7,10,21,11,22,12,33)(6,13,16,36,17,37,18,44)(8,19,25,47,26,48,27,53)(9,20,32,54,30,51,31,52)(14,34,23,45,39,60,40,63)(15,35,28,58,42,62,43,49)(24,46,57,67,29,50,56,66)(38,59,55,65,41,61,64,68);(2,4)(5,8,11,26)(6,14,17,39)(7,19,22,48)(9,29,30,24)(10,27,12,25)(13,34,37,60)(15,41,42,38)(16,40,18,23)(20,50,51,46)(21,53,33,47)(28,55,43,64)(31,56,32,57)(35,61,62,59)(36,63,44,45)(49,68,58,65)(52,66,54,67);(5,9,10,32,11,30,12,31)(6,15,16,28,17,42,18,43)(7,20,21,54,22,51,33,52)(8,24,25,57,26,29,27,56)(13,35,36,58,37,62,44,49)(14,38,23,55,39,41,40,64)(19,46,47,67,48,50,53,66)(34,59,45,65,60,61,63,68)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380327);