Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213778 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1544 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)(7,15,18,11)(8,19,21,30)(9,14)(10,13)(12,22,28,23)(16,33)(17,31,24,27)(20,36)(25,29)(26,34)(32,35);(1,2,3,4)(5,7)(6,11)(8,20)(9,17)(10,18)(12,26)(13,15)(14,27)(16,21)(19,36)(22,35)(23,34)(24,25)(28,32)(29,31)(30,33);(2,4)(5,8,9,23,10,21,25,22)(6,12,14,30,13,28,29,19)(7,16,24,34,18,20,17,35)(11,26,31,36,15,32,27,33);(5,7)(6,11)(8,16)(9,24)(10,18)(12,26)(13,15)(14,31)(17,25)(19,33)(20,21)(22,35)(23,34)(27,29)(28,32)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=fh,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282394);