Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3898 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)(7,15,18,11)(8,19,20,30)(9,14)(10,13)(12,21,27,23)(16,32)(17,29,22,26)(24,28)(25,34)(31,35)(33,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,22)(10,18)(12,25)(13,15)(14,29)(17,24)(19,32)(20,33)(21,35)(23,34)(26,28)(27,31)(30,36);(2,4)(5,8)(6,12)(7,16)(9,23)(10,20)(11,25)(13,27)(14,30)(15,31)(17,35)(18,33)(19,28)(21,24)(22,34)(26,32)(29,36);(5,7)(6,11)(8,16)(9,22)(10,18)(12,25)(13,15)(14,29)(17,24)(19,32)(20,33)(21,35)(23,34)(26,28)(27,31)(30,36)Polycyclic presentation:
2^9|a^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72316856 |
| maximal normal subgroup of | #72316856 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266734);