Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196248 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1878 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)(7,15)(8,19,21,13)(9,22)(10,12)(11,17)(14,24)(16,30,32,36)(18,29)(20,27,23,28)(25,34)(26,31,35,33);(5,7,9,17,10,18,24,34)(6,11,14,15,12,25,22,29)(8,16,23,33,21,32,20,31)(13,26,28,30,19,35,27,36);(1,2,3,4)(6,12)(11,25)(13,19)(14,22)(15,29)(26,35)(27,28)(30,36);(5,8)(6,13)(7,16)(9,23)(10,21)(11,26)(12,19)(14,28)(15,30)(17,33)(18,32)(20,24)(22,27)(25,35)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,g^b=gi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300218);