Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,9)(8,19)(10,18)(11,14)(12,16)(13,22)(15,28)(17,33)(20,36)(21,31)(23,35)(24,34)(25,32)(26,29)(27,30);(1,2)(3,4)(5,7,11,16,12,14,9,6)(8,20,21,17,24,13,23,30)(10,26,28,15,29,18,25,32)(19,33,35,36,34,27,31,22);(2,4)(5,8,9,21,12,24,11,23)(6,13,14,30,16,20,7,17)(10,27,25,22,29,33,28,36)(15,31,26,19,32,35,18,34);(5,9,12,11)(6,7,16,14)(8,21,24,23)(10,25,29,28)(13,17,20,30)(15,18,32,26)(19,31,34,35)(22,33,36,27)Polycyclic presentation:
2^9|b^2=h,c^2=e,d^2=i,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231480, #72231484, #72231495, … |
| maximal normal subgroup of | #72231480, #72231484, #72231495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406252);