Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,10,32,12,16,11,17)(7,13,33,37,20,39,21,38)(8,22,25,14,26,43,34,42)(9,27,29,15,31,45,30,44)(18,46,49,40,23,53,48,35)(19,50,52,36,28,60,51,63)(24,56,57,61,58,41,59,54)(47,62,67,68,55,64,66,65);(1,2)(3,4)(5,7,11,33,12,20,10,21)(6,13,32,38,16,39,17,37)(8,23,34,48,26,18,25,49)(9,28,30,51,31,19,29,52)(14,40,43,46,42,35,22,53)(15,36,45,50,44,63,27,60)(24,47,59,67,58,55,57,66)(41,64,54,68,56,62,61,65);(2,4)(5,8)(6,14,16,42)(7,18,20,23)(9,24)(10,25)(11,34)(12,26)(13,35)(15,41,44,56)(17,22,32,43)(19,47,28,55)(21,48,33,49)(27,61,45,54)(29,57)(30,59)(31,58)(36,62)(37,46)(38,53)(39,40)(50,65)(51,66,52,67)(60,68)(63,64);(5,9,12,31)(6,15,16,44)(7,19,20,28)(8,24,26,58)(10,30,11,29)(13,36,39,63)(14,41,42,56)(17,45,32,27)(18,47,23,55)(21,52,33,51)(22,54,43,61)(25,59,34,57)(35,62,40,64)(37,50,38,60)(46,65,53,68)(48,67,49,66)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,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, 329102);