Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499757 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,13)(7,17)(8,21)(9,22)(10,11)(14,30)(15,23)(16,25)(18,35)(19,24)(20,28)(26,27)(29,32)(31,33);(1,2)(3,4)(5,6,12,8)(7,14,20,18)(9,23,27,25)(10,16,11,15)(13,26,21,22)(17,33,28,32)(19,35,24,30)(29,34,31,36);(2,4)(5,7)(6,14)(8,18)(9,24)(10,28)(11,17)(12,20)(13,29)(15,33)(16,32)(19,27)(21,31)(22,36)(23,30)(25,35)(26,34);(5,8,12,6)(7,18,20,14)(9,25,27,23)(10,15,11,16)(13,22,21,26)(17,32,28,33)(19,30,24,35)(29,36,31,34)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250402 |
| maximal normal subgroup of | #72250402 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406957);