Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232138 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,18)(8,14,25,13)(9,26,27,33)(10,17)(11,16)(12,21)(15,24)(19,32,35,34)(20,29,23,36)(22,31)(28,30);(1,2)(3,4)(5,7)(6,12)(8,23)(9,19)(10,28)(11,22)(13,32)(14,34)(15,31)(16,30)(17,18)(20,25)(21,24)(26,36)(27,35)(29,33);(5,8,24,27,11,25,10,9)(6,13,15,26,16,14,17,33)(7,19,28,20,22,35,21,23)(12,29,18,34,30,36,31,32);(1,3,4,2)(5,9,10,25,11,27,24,8)(6,14,15,33,16,13,17,26)(7,20,21,19,22,23,28,35)(12,29,18,34,30,36,31,32)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250742, #72250758, #72250789, … |
| maximal normal subgroup of | #72250742, #72250758, #72250789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293230);