Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1926 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)(7,17,20,12)(8,21)(9,16)(10,15)(11,14)(13,23)(18,26,25,30)(19,33,35,36)(22,28)(24,29)(27,34,31,32);(1,2)(3,4)(7,18,20,25)(12,26,17,30)(19,34,35,32)(27,36,31,33);(2,4)(5,7)(6,12)(8,19)(9,25)(10,18)(11,20)(13,27)(14,17)(15,30)(16,26)(21,36)(22,32)(23,34)(24,35)(28,31)(29,33);(5,8,11,24)(6,13,14,28)(7,19,20,35)(9,23,10,22)(12,27,17,31)(15,21,16,29)(18,32,25,34)(26,33,30,36)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325100);