Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(22,28)(23,33)(25,26)(30,34)(31,32)(35,36);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,22,26,31,27,32,25,28)(23,29,33,35,34,24,30,36);(2,4)(7,13)(10,16)(14,19)(17,20)(21,23,25,30,27,34,26,33)(22,29,28,36,32,24,31,35);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,24,27,29)(22,30,32,33)(23,28,34,31)(25,35,26,36);(8,15)(11,18)(14,19)(17,20)(23,34)(24,29)(30,33)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473385);