Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
5696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(11,15)(13,16)(17,18,21,19)(20,23,24,22);(1,2)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(18,19)(22,23);(2,6)(4,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,18)(20,23,24,22);(2,6)(3,8)(17,20)(18,22)(19,23)(21,24);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,15)(14,16)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840069);