Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1520 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)(6,12)(8,21)(9,10)(13,26)(15,25)(16,29)(17,34)(18,19)(22,35)(23,24)(27,31)(28,36);(5,6,9,12,11,15,10,25)(7,16,18,29,20,26,19,13)(8,14,21,27,24,32,23,31)(17,33,34,36,35,30,22,28);(2,4)(5,7,11,20)(6,13,15,29)(8,22,24,34)(9,19,10,18)(12,26,25,16)(14,30,32,33)(17,21,35,23)(27,36,31,28);(5,8)(6,14)(7,17)(9,21)(10,23)(11,24)(12,27)(13,28)(15,32)(16,33)(18,34)(19,22)(20,35)(25,31)(26,30)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371857);