Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563346 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,19)(8,15)(9,28)(10,17)(11,16)(12,18)(13,23)(14,26)(20,30)(21,31)(22,35)(24,33)(25,32)(27,34)(29,36);(1,2)(3,4)(5,7,10,23,12,25,11,24)(6,13,16,19,18,33,17,32)(8,22,26,20,27,21,9,29)(14,31,15,30,28,35,34,36);(5,8,11,9,12,27,10,26)(6,14,17,34,18,28,16,15)(7,20,23,21,25,29,24,22)(13,30,19,35,33,36,32,31);(2,4)(5,9,10,8,12,26,11,27)(6,15,16,28,18,34,17,14)(7,21,24,20,25,22,23,29)(13,31,32,36,33,35,19,30);(5,8,10,26,12,27,11,9)(6,14,16,15,18,28,17,34)(7,22,23,20,25,21,24,29)(13,31,19,30,33,35,32,36)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=g,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470544);