Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2537 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12,11,16,10,15)(7,17,20,26,21,13,18,29)(8,22,24,14,23,27,25,30)(19,33,34,28,35,36,32,31);(1,2,3,4)(6,12,16,15)(7,18,21,20)(8,23)(13,17)(14,22,30,27)(19,34,35,32)(24,25)(26,29);(2,4)(5,7)(6,13)(8,19)(9,18)(10,20)(11,21)(12,26)(14,28)(15,29)(16,17)(22,36)(23,35)(24,32)(25,34)(27,33)(30,31);(5,8,9,24,11,23,10,25)(6,14,15,22,16,30,12,27)(7,19,18,32,21,35,20,34)(13,28,29,36,17,31,26,33)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269243);