Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1614 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)(7,16)(8,20)(9,14)(10,24)(11,15)(12,18)(13,23)(17,31)(19,27)(21,35)(22,29)(25,28)(26,36)(30,33)(32,34);(5,7)(6,12)(8,17,23,33)(9,18)(10,25)(11,19)(13,26,20,34)(14,27)(15,28)(16,24)(21,32,22,36)(29,31,35,30);(5,8,10,22,11,23,9,21)(6,13,14,29,15,20,24,35)(7,17,18,32,19,33,25,36)(12,26,16,30,28,34,27,31);(5,9,11,10)(6,14,15,24)(7,18,19,25)(8,21,23,22)(12,27,28,16)(13,29,20,35)(17,32,33,36)(26,31,34,30)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265496);