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 | |
8062 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)(7,11)(8,10)(9,12)(15,21)(19,22)(23,27)(26,28);(1,2,3,4)(13,14)(15,22)(16,24)(17,18)(19,21)(20,25)(23,26)(27,28);(5,6,8,10)(7,9,12,11)(14,17)(19,22)(20,24)(26,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,28)(25,27);(5,7,8,12)(6,9,10,11)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28);(13,17,18,14)(15,22,21,19)(16,24,25,20)(23,28,27,26)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7718154);