Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394823 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1573 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)(7,17)(8,11)(9,23)(10,15)(12,16)(13,19)(14,25)(18,27)(20,29)(21,28)(22,26)(24,30)(31,33)(32,35)(34,36);(5,7,11,20,12,21,10,19)(6,13,15,28,16,29,8,17)(9,18,25,33,26,34,24,32)(14,27,23,35,30,36,22,31);(2,4)(5,8,10,16,12,15,11,6)(7,13,19,17,21,29,20,28)(9,22,24,30,26,23,25,14)(18,27,32,31,34,36,33,35);(5,9)(6,14)(7,18)(8,22)(10,24)(11,25)(12,26)(13,27)(15,23)(16,30)(17,31)(19,32)(20,33)(21,34)(28,35)(29,36)Polycyclic presentation:
2^9|b^2=ei,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,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, 302020);