Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393921 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62848 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,28,11,16,12,15)(7,17,21,8,20,23,19,13)(9,24,25,14,26,31,27,30)(18,29,35,36,34,22,33,32);(1,2)(3,4)(5,7)(6,13)(8,16)(9,18)(10,21)(11,20)(12,19)(14,29)(15,23)(17,28)(22,30)(24,32)(25,35)(26,34)(27,33)(31,36);(2,4)(5,8)(6,7)(9,22)(10,17)(11,13)(12,23)(14,18)(15,21)(16,20)(19,28)(24,35)(25,36)(26,29)(27,32)(30,34)(31,33);(5,9,11,26)(6,14,16,30)(7,18,20,34)(8,22,13,29)(10,27,12,25)(15,31,28,24)(17,32,23,36)(19,35,21,33)Polycyclic presentation:
2^9|a^2=ef,d^2=i,e^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,e^c=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302595);