Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | #2495 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,11,18,10,17)(7,19,22,29,21,15,9,13)(8,14,25,33,26,34,27,32)(20,30,24,28,35,31,23,36);(1,2)(3,4)(5,7,10,9,11,21,12,22)(6,13,16,19,18,29,17,15)(8,23,27,35,26,24,25,20)(14,30,33,28,34,31,32,36);(2,4)(5,8)(6,14)(7,20)(9,24)(10,25)(11,26)(12,27)(13,28)(15,31)(16,32)(17,33)(18,34)(19,30)(21,35)(22,23)(29,36);(5,9,10,21,11,22,12,7)(6,15,16,13,18,19,17,29)(8,24,25,35,26,23,27,20)(14,31,32,28,34,30,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=fh,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, 272445);