Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,9,16,11,14)(7,12,20,27,18,29,19,28)(8,21,24,31,23,32,25,13)(17,33,22,26,34,30,35,36);(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,30)(14,27)(15,28)(16,29)(17,24)(21,26)(23,35)(25,34)(31,33)(32,36);(2,4)(5,8)(6,13)(7,17)(9,23)(10,25)(11,24)(12,26)(14,21)(15,32)(16,31)(18,34)(19,22)(20,35)(27,33)(28,30)(29,36);(6,14,16,15)(8,23)(12,27,29,28)(13,21,31,32)(17,34)(22,35)(24,25)(26,33,36,30)Polycyclic presentation:
2^9|a^2=g,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319622, #72319923, #72320018 |
| maximal normal subgroup of | #72319622, #72319923, #72320018 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367504);