Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1536 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,28)(20,31)(21,25)(22,27)(23,26)(24,29)(30,32);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,29)(18,24,25,28)(20,22,30,23)(26,32,27,31);(10,13)(14,15)(17,20,22,29,21,30,23,19)(18,24,26,31,25,28,27,32);(9,11,12,16)(10,14,13,15)(17,20,23,19,21,30,22,29)(18,24,27,32,25,28,26,31);(1,3,4,8)(2,6,5,7)(18,25)(24,28)(26,27)(31,32)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=gi,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=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503620);