Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| 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 | |
2091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,17)(7,18,21,28)(8,22,26,14)(9,20,13,29)(10,16,11,15)(19,34,23,33)(24,36,30,35)(25,31,27,32);(5,7)(6,13)(8,23)(9,17)(10,29)(11,20)(12,21)(14,24)(15,28)(16,18)(19,26)(22,30)(25,35)(27,36)(31,33)(32,34);(2,4)(5,8,10,25,12,26,11,27)(6,14,15,31,17,22,16,32)(7,19,20,35,21,23,29,36)(9,24,28,34,13,30,18,33);(5,9,12,13)(6,7,17,21)(8,24,26,30)(10,28,11,18)(14,19,22,23)(15,20,16,29)(25,34,27,33)(31,35,32,36)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301875);