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 | |
7531 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)(8,9)(11,12)(13,14,19,22)(15,17,20,16)(18,21,26,28)(23,25,27,24);(1,2,3,4)(13,15,19,20)(14,16,22,17)(18,23,26,27)(21,24,28,25);(6,9)(10,12)(13,16,19,17)(14,20,22,15)(18,25,26,24)(21,23,28,27);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,20,22,15)(18,24,26,25)(21,23,28,27);(5,7)(6,10)(8,11)(9,12)(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,27)(22,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982860);