Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 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)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(19,20)(23,32)(24,28)(29,30);(1,3,4,8)(2,5,6,7)(17,18)(19,28)(20,24)(21,26)(22,25)(23,29)(27,31)(30,32);(10,13)(14,16)(17,19,21,20)(18,23,26,32)(22,30,31,29)(24,25,28,27);(2,6)(5,7)(17,20,21,19)(18,24,26,28)(22,30,31,29)(23,25,32,27);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(22,31)(23,28)(24,32)(26,27)(29,30)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6618791);