Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455155 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23409 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,18,10,17)(7,19,23,32,24,31,22,13)(8,25,21,34,28,33,27,14)(9,15,20,30,26,36,29,35);(1,2)(3,4)(5,7,10,22,12,24,11,23)(6,13,16,19,18,32,17,31)(8,26,27,20,28,9,21,29)(14,15,25,30,34,36,33,35);(2,4)(5,8)(6,14)(7,20)(9,23)(10,21)(11,27)(12,28)(13,30)(15,19)(16,33)(17,25)(18,34)(22,26)(24,29)(31,36)(32,35);(5,9,10,29,12,26,11,20)(6,15,16,30,18,36,17,35)(7,21,22,8,24,27,23,28)(13,25,19,34,32,33,31,14)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362351);