Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | |
1273 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)(7,18)(8,20,25,32)(9,19,28,15)(10,29)(11,16)(12,17)(13,21)(14,24,33,30)(22,31)(23,35)(26,34,27,36);(1,2)(3,4)(5,7,12,23)(6,13,17,31)(8,15,25,19)(9,14,28,33)(10,22,11,21)(16,35,29,18)(20,27,32,26)(24,36,30,34);(2,4)(5,8,10,30,12,25,11,24)(6,14,16,20,17,33,29,32)(7,19,21,34,23,15,22,36)(9,18,27,31,28,35,26,13);(5,9)(6,15)(7,20)(8,18)(10,27)(11,26)(12,28)(13,24)(14,22)(16,36)(17,19)(21,33)(23,32)(25,35)(29,34)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295581);