Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6)(7,18)(8,24)(9,15)(10,32)(11,16)(12,17)(13,21)(14,27)(19,47)(20,45)(22,36)(23,46)(25,34)(26,55)(28,39)(29,43)(30,41)(31,42)(33,40)(35,52)(37,49)(38,57)(44,50)(48,67)(51,64)(53,62)(54,63)(56,60)(58,65)(59,66)(61,68);(5,7,10,22,12,21,11,23)(6,13,16,36,17,18,32,46)(8,25,27,19,29,50,28,49)(9,20,30,53,31,52,33,54)(14,37,24,44,39,47,43,34)(15,35,40,62,42,45,41,63)(26,56,57,48,58,68,59,51)(38,64,55,61,66,67,65,60);(5,8,12,29)(6,14,17,39)(7,19,21,49)(9,26,31,58)(10,28,11,27)(13,34,18,44)(15,38,42,66)(16,43,32,24)(20,51,52,48)(22,25,23,50)(30,59,33,57)(35,61,45,60)(36,47,46,37)(40,65,41,55)(53,68,54,56)(62,64,63,67);(1,3,4,2)(5,9,12,31)(6,15,17,42)(7,20,21,52)(8,26,29,58)(10,30,11,33)(13,35,18,45)(14,38,39,66)(16,40,32,41)(19,48,49,51)(22,53,23,54)(24,55,43,65)(25,56,50,68)(27,57,28,59)(34,60,44,61)(36,62,46,63)(37,64,47,67)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,d^c=de,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288811);