Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500733 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,21)(8,27)(9,19)(10,32)(11,16)(12,22)(13,20)(14,24)(15,28)(17,33)(18,31)(23,35)(25,36)(26,30)(29,34);(5,7,10,24,13,27,33,26)(6,14,17,21,20,30,32,8)(9,23,11,34,31,25,12,15)(16,36,19,29,22,35,18,28);(2,4)(5,8)(6,15)(7,22)(9,29)(10,14)(11,35)(12,36)(13,21)(16,27)(17,23)(18,24)(19,26)(20,34)(25,32)(28,31)(30,33);(5,9,10,11,13,31,33,12)(6,16,17,19,20,22,32,18)(7,23,24,34,27,25,26,15)(8,28,14,36,21,29,30,35)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407929);