Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419919 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,12,16,11,18,10,17)(7,19,33,36,23,37,22,13)(8,14,25,45,26,41,27,40)(9,28,32,15,31,43,30,44)(20,48,50,38,52,46,51,34)(21,53,55,47,54,35,29,42)(24,56,58,39,60,61,59,65)(49,63,57,66,67,62,68,64);(5,7,10,22,11,23,12,33)(6,13,16,19,18,36,17,37)(8,20,25,50,26,52,27,51)(9,29,30,54,31,55,32,21)(14,38,40,48,41,34,45,46)(15,42,43,53,44,47,28,35)(24,57,58,67,60,68,59,49)(39,64,56,62,65,66,61,63);(5,8,10,25,11,26,12,27)(6,14,17,45,18,41,16,40)(7,20,33,51,23,52,22,50)(9,24,32,59,31,60,30,58)(13,34,19,46,36,38,37,48)(15,39,43,56,44,65,28,61)(21,49,29,57,54,67,55,68)(35,62,47,64,53,63,42,66);(2,4)(5,9,11,31)(6,15,18,44)(7,21,23,54)(8,24,26,60)(10,32,12,30)(13,35,36,53)(14,39,41,65)(16,28,17,43)(19,47,37,42)(20,49,52,67)(22,55,33,29)(25,59,27,58)(34,62,38,63)(40,61,45,56)(46,64,48,66)(50,68,51,57)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=efi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341274);