Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,12,16,11,7,10,15)(8,13,22,29,21,17,20,28)(9,23,26,14,25,30,24,18)(19,32,35,31,34,36,33,27);(1,2,3,4)(5,7,11,6)(8,17,21,13)(9,18,25,14)(10,15,12,16)(19,31,34,27)(20,28,22,29)(23,26,30,24)(32,33,36,35);(5,8)(6,13)(7,17)(9,19)(10,22)(11,21)(12,20)(14,27)(15,29)(16,28)(18,31)(23,32)(24,35)(25,34)(26,33)(30,36);(5,9,11,25)(6,14,7,18)(8,19,21,34)(10,26,12,24)(13,27,17,31)(15,30,16,23)(20,35,22,33)(28,32,29,36)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299922);