Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476175 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15,9,16,10,13)(7,12,20,28,18,29,19,26)(8,21,25,14,23,30,24,32)(17,31,22,35,33,36,34,27);(1,2)(3,4)(5,7)(6,12)(8,22)(9,18)(10,20)(11,19)(13,28)(14,31)(15,26)(16,29)(17,25)(21,35)(23,34)(24,33)(27,30)(32,36);(2,4)(6,13,16,15)(8,23)(12,26,29,28)(14,30,32,21)(17,33)(22,34)(24,25)(27,31,35,36);(5,8,9,23)(6,14,16,32)(7,17,18,33)(10,25,11,24)(12,27,29,35)(13,30,15,21)(19,22,20,34)(26,31,28,36)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,d^c=de,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285454, #72285507, #72285613 |
| maximal normal subgroup of | #72285454, #72285507, #72285613 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383720);