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 | |
7067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(5,6,9,8)(7,11,12,10)(13,14,18,20)(15,19,23,27)(16,21,25,28)(17,22,24,26);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,27)(21,26)(22,28);(2,3)(5,6,9,8)(7,10,12,11)(14,20)(15,23)(16,25)(22,26);(1,3,4,2)(13,16,18,25)(14,21,20,28)(15,24,23,17)(19,26,27,22);(5,6,9,8)(7,10,12,11)(13,17,18,24)(14,22,20,26)(15,16,23,25)(19,21,27,28);(5,8,9,6)(7,11,12,10)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7590081);