Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394823 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| 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 | |
1176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6)(7,19)(8,25)(9,31)(10,17)(11,16)(12,18)(13,22)(14,28)(15,32)(20,47)(21,51)(23,38)(24,37)(26,45)(27,57)(29,42)(30,41)(33,43)(34,44)(35,50)(36,54)(39,49)(40,60)(46,52)(48,63)(53,62)(55,58)(56,66)(59,64)(61,68)(65,67);(5,7,10,22,12,24,11,23)(6,13,16,19,18,38,17,37)(8,26,28,50,30,49,29,20)(9,21,32,52,33,54,34,53)(14,39,25,35,42,45,41,47)(15,36,31,46,44,51,43,62)(27,56,58,67,60,68,59,48)(40,63,55,61,57,65,64,66);(1,2,4,3)(5,8,10,28,12,30,11,29)(6,14,16,25,18,42,17,41)(7,20,22,26,24,50,23,49)(9,27,32,58,33,60,34,59)(13,35,19,45,38,47,37,39)(15,40,31,55,44,57,43,64)(21,48,52,56,54,67,53,68)(36,61,46,65,51,66,62,63);(1,3,4,2)(5,9,12,33)(6,15,18,44)(7,21,24,54)(8,27,30,60)(10,32,11,34)(13,36,38,51)(14,40,42,57)(16,31,17,43)(19,46,37,62)(20,48,50,67)(22,52,23,53)(25,55,41,64)(26,56,49,68)(28,58,29,59)(35,61,47,66)(39,63,45,65)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308481);