Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1047 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)(5,6,12,16,10,9,11,15)(7,13,20,29,19,18,27,30)(8,21,25,14,26,32,24,23)(17,31,34,33,35,36,22,28);(5,7)(6,13)(8,22)(9,18)(10,19)(11,20)(12,27)(14,31)(15,29)(16,30)(17,24)(21,33)(23,36)(25,35)(26,34)(28,32);(5,8,10,26)(6,14,9,23)(7,17)(11,25,12,24)(13,28)(15,32,16,21)(18,33)(19,35)(20,22)(27,34)(29,36)(30,31);(5,9)(6,10)(7,18,19,13)(8,23)(11,16)(12,15)(14,26)(17,33,35,28)(20,30,27,29)(21,25)(22,31,34,36)(24,32)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=efgi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295281);