Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456501 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,16,11,17,10,18)(7,19,24,36,23,45,22,13)(8,14,32,39,27,40,26,41)(9,28,33,15,31,43,30,44)(20,46,51,34,50,61,49,37)(21,35,55,63,54,64,53,47)(25,56,58,38,57,42,29,59)(48,65,68,66,67,60,52,62);(1,2)(3,4)(5,7,11,23)(6,13)(8,20,27,50)(9,21,31,54)(10,22,12,24)(14,37)(15,35)(16,19)(17,36)(18,45)(25,48,57,67)(26,49,32,51)(28,47)(29,52,58,68)(30,53,33,55)(34,40)(38,65)(39,46)(41,61)(42,66)(43,63)(44,64)(56,62)(59,60);(2,4)(5,8,11,27)(6,14,17,40)(7,20,23,50)(9,29,31,58)(10,32,12,26)(13,34,36,37)(15,42,44,56)(16,41,18,39)(19,46,45,61)(21,52,54,68)(22,51,24,49)(25,33,57,30)(28,59,43,38)(35,62,64,66)(47,65,63,60)(48,55,67,53);(5,9,11,31)(6,15,17,44)(7,21,23,54)(8,25,27,57)(10,33,12,30)(13,35,36,64)(14,38,40,59)(16,28,18,43)(19,47,45,63)(20,48,50,67)(22,55,24,53)(26,58,32,29)(34,60,37,65)(39,56,41,42)(46,62,61,66)(49,68,51,52)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=i,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,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363700);