Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396611 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1205 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)(7,16)(8,20)(9,25)(10,14)(11,15)(12,18)(13,23)(17,30)(19,28)(21,35)(22,31)(24,34)(26,29)(27,36)(32,33);(5,7)(6,12)(8,21,23,36)(9,26)(10,18)(11,19)(13,30,20,32)(14,28)(15,29)(16,25)(17,24,33,22)(27,31,35,34);(5,8,9,22,11,23,10,24)(6,13,14,31,15,20,25,34)(7,17,18,21,19,33,26,36)(12,27,16,32,29,35,28,30);(2,4)Polycyclic presentation:
2^9|b^2=f,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303807);