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 | |
7443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,20)(16,17)(18,23)(19,21)(22,28)(24,25)(26,27);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,21,23)(16,24,22,26)(17,25,28,27);(5,7,9,8)(6,10,12,11)(14,21)(16,22)(20,23)(24,26);(5,7,9,8)(6,11,12,10)(13,16,19,22)(14,17,21,28)(15,24,18,26)(20,25,23,27);(5,6)(7,11)(8,10)(9,12)(13,17)(14,22)(15,25)(16,21)(18,27)(19,28)(20,26)(23,24);(5,8,9,7)(6,11,12,10)(13,18,19,15)(14,23,21,20)(16,26,22,24)(17,27,28,25)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894170);