Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,7,4,6)(17,18,22,25)(19,26,21,23)(20,28,29,24)(27,31,30,32);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32);(1,3,5,4)(2,7,8,6)(10,14)(11,12)(17,20,22,29)(18,24,25,28)(19,27,21,30)(23,31,26,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,21,22,19)(18,23,25,26)(20,27,29,30)(24,32,28,31);(9,12,13,11)(10,15,14,16)(18,25)(23,26)(24,28)(31,32);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615317);