Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,4,7,8)(10,13)(14,16)(17,18,23,20)(19,21,25,26)(22,24,30,28)(27,29,31,32);(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31);(2,6)(3,7)(18,20)(19,25)(22,30)(29,32);(1,3)(2,4)(5,7)(6,8)(17,20,23,18)(19,26,25,21)(22,28,30,24)(27,32,31,29);(1,4)(2,3)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,21,23,26)(18,19,20,25)(22,29,30,32)(24,27,28,31);(17,22)(18,24)(19,27)(20,28)(21,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=i,e^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7703019);