Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| 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 | |
1834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,18)(8,22)(9,24,27,30)(10,16)(11,15)(12,17)(13,20)(14,26,31,25)(19,33,23,28)(21,29)(32,35,36,34);(1,2)(3,4)(5,7,10,20,12,8,11,21)(6,13,15,18,17,29,16,22)(9,19,25,34,27,23,26,35)(14,28,24,32,31,33,30,36);(2,4)(5,8,12,7)(6,13,17,29)(9,23,27,19)(10,20,11,21)(14,28,31,33)(15,22,16,18)(24,36,30,32)(25,34,26,35);(5,9)(6,14)(7,19)(8,23)(10,25)(11,26)(12,27)(13,28)(15,24)(16,30)(17,31)(18,32)(20,34)(21,35)(22,36)(29,33)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302930);