Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25770 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,19)(20,27,30,31)(21,28,25,22)(24,32,26,29);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,18)(20,27,30,31)(21,22,25,28)(24,32,26,29);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,28,32,23,30,22,29)(18,24,21,31,19,26,25,27);(9,11,12,16)(10,14,13,15)(17,21,22,18,23,25,28,19)(20,27,29,26,30,31,32,24);(1,4,5,3)(2,7,6,8)(17,22,23,28)(18,21,19,25)(20,29,30,32)(24,31,26,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504640);