Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1444 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)(5,6,10,15,9,16,11,13)(7,12,17,26,19,27,20,30)(8,21,23,28,22,14,24,29)(18,32,31,36,33,25,34,35);(5,7)(6,12)(8,18)(9,19)(10,20)(11,17)(13,26)(14,25)(15,30)(16,27)(21,32)(22,33)(23,34)(24,31)(28,35)(29,36);(2,4)(6,13)(7,17)(10,11)(14,28)(15,16)(18,31)(19,20)(21,29)(23,24)(26,30)(33,34)(35,36);(5,8,9,22)(6,14)(7,18,19,33)(10,23,11,24)(12,25)(13,28)(15,29)(16,21)(17,31,20,34)(26,35)(27,32)(30,36)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303309);