Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454676 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,18)(8,22)(9,24,27,33)(10,16)(11,15)(12,17)(13,20)(14,28,32,26)(19,29,23,36)(21,30)(25,31,35,34);(5,7,12,8)(6,13,17,30)(9,19,27,23)(10,21,11,20)(14,29,32,36)(15,22,16,18)(24,34,33,31)(25,28,35,26);(2,4)(5,8,12,7)(6,13,17,30)(9,25,27,35)(10,20,11,21)(14,31,32,34)(15,22,16,18)(19,26,23,28)(24,29,33,36);(5,9)(6,14)(7,19)(8,23)(10,28)(11,26)(12,27)(13,29)(15,33)(16,24)(17,32)(18,34)(20,25)(21,35)(22,31)(30,36)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=i,e^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,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361875);