Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,15,12,16,11,26)(7,13,18,29,19,30,20,35)(8,21,23,31,9,24,25,14)(17,32,33,28,22,36,34,27);(1,2)(3,4)(5,7,10,18,12,19,11,20)(6,13,15,29,16,30,26,35)(8,22,23,34,9,17,25,33)(14,28,21,36,31,27,24,32);(1,2,4,3)(5,8,12,9)(6,14,16,31)(7,17,19,22)(10,25,11,23)(13,27,30,28)(15,24,26,21)(18,34,20,33)(29,36,35,32);(1,3,4,2)(5,9,12,8)(6,14,16,31)(7,17,19,22)(10,25,11,23)(13,28,30,27)(15,21,26,24)(18,33,20,34)(29,36,35,32)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=ei,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329880);