Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| 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 | #2537 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,15)(7,16,20,27)(8,21,23,25)(9,11,24,14)(12,19,28,17)(13,29,30,22)(18,26,34,36)(31,32,35,33);(1,3,4,2)(6,11,15,14)(7,17,20,19)(13,25,30,21)(18,32,34,33);(5,7)(6,12)(8,18)(9,19)(10,20)(11,16)(13,26)(14,27)(15,28)(17,24)(21,35)(22,33)(23,34)(25,31)(29,32)(30,36);(5,8,9,22,10,23,24,29)(6,13,14,21,15,30,11,25)(7,18,19,33,20,34,17,32)(12,26,27,35,28,36,16,31)Polycyclic presentation:
2^9|a^2=h,b^2=egh,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273722);