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 | |
7335 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,27)(18,22)(21,25)(23,28)(24,26);(1,2,3,4)(13,15)(14,19)(16,25)(17,24)(18,23)(20,21)(22,28)(26,27);(5,6,8,10)(7,9,11,12)(15,23)(17,25)(19,28)(21,27);(5,7,8,11)(6,9,10,12)(13,16)(14,20)(15,17)(18,24)(19,27)(21,28)(22,26)(23,25);(5,7)(6,9)(8,11)(10,12)(13,17)(14,21)(15,24)(16,23)(18,25)(19,20)(22,27)(26,28);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,26)(17,21)(18,22)(20,24)(23,28)(25,27)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7985028);