Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16,12,15,10,7)(8,19,22,29,23,17,21,13)(9,24,26,30,25,14,27,18)(20,31,34,36,33,32,35,28);(1,2,3,4)(5,7,11,15,12,16,10,6)(8,17,22,29,23,19,21,13)(9,18,27,24,25,30,26,14)(20,31,35,36,33,32,34,28);(5,8,11,22,12,23,10,21)(6,13,7,17,15,29,16,19)(9,20,27,35,25,33,26,34)(14,28,18,31,24,36,30,32);(5,9)(6,14)(7,18)(8,20)(10,26)(11,27)(12,25)(13,28)(15,24)(16,30)(17,31)(19,32)(21,34)(22,35)(23,33)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,e^b=ei,g^b=gi,h^b=hi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312242, #72312257, #72312265, … |
| maximal normal subgroup of | #72312242, #72312257, #72312265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300231);