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 | |
4946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,13)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(30,31);(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,29)(21,31)(22,28)(24,25)(26,27)(30,32);(1,3,5,8)(2,6,7,4)(19,28)(21,29)(23,27)(25,32);(1,3)(2,4)(5,8)(6,7)(17,20)(18,24)(19,29)(21,28)(22,31)(23,25)(26,30)(27,32);(1,4)(2,3)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,31)(27,28)(29,32);(17,21,22,29)(18,25,26,32)(19,20,28,31)(23,30,27,24)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7754589);