Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,32)(30,31);(1,3,5,4)(2,6,8,7)(18,23)(19,20)(22,28)(24,25)(27,31)(29,30);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,26)(31,32);(1,4,5,3)(2,6,8,7)(17,20,21,30)(18,25,23,31)(19,28,29,22)(24,26,27,32);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364076);