Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,8)(9,11)(10,12)(13,14,18,21)(15,16,19,23)(17,25,26,20)(22,28,27,24);(1,2)(3,4)(5,7,10,9)(6,8,12,11)(15,19)(16,23)(22,27)(24,28);(5,7,10,9)(6,11,12,8)(13,15)(14,16)(17,22)(18,19)(20,24)(21,23)(25,28)(26,27);(5,8,10,11)(6,7,12,9);(5,7,10,9)(6,8,12,11)(13,16)(14,19)(15,21)(17,24)(18,23)(20,27)(22,25)(26,28);(5,9,10,7)(6,11,12,8)(13,17,18,26)(14,20,21,25)(15,22,19,27)(16,24,23,28)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7587833);