Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,15,9,13)(7,17,19,28,20,12,24,26)(8,21,22,30,23,31,25,14)(18,32,34,27,35,36,33,29);(1,2)(3,4)(5,7,9,19,11,20,10,24)(6,12,13,26,15,17,16,28)(8,18,22,33,23,35,25,34)(14,29,21,36,30,27,31,32);(2,4)(6,13,15,16)(12,26,17,28)(14,21,30,31)(27,32,29,36);(5,8)(6,14)(7,18)(9,22)(10,25)(11,23)(12,27)(13,21)(15,30)(16,31)(17,29)(19,33)(20,35)(24,34)(26,32)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320778 |
| maximal normal subgroup of | #72320778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357163);