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 | |
6960 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)(5,6,9,11)(7,8,12,10)(13,14)(15,20)(16,22)(17,26)(18,23)(19,21)(24,27)(25,28);(1,2,3,4)(13,15,19,24)(14,20,21,27)(16,23,26,28)(17,25,22,18);(5,7,9,12)(6,10,11,8)(14,21)(15,24)(16,26)(18,25);(5,7)(6,8)(9,12)(10,11)(13,16,19,26)(14,22,21,17)(15,23,24,28)(18,27,25,20);(5,8)(6,7)(9,10)(11,12)(13,17,19,22)(14,16,21,26)(15,25,24,18)(20,23,27,28);(13,18,19,25)(14,23,21,28)(15,22,24,17)(16,27,26,20)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7756877);