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 | |
6380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,15)(11,12,13,16)(17,18)(19,20)(21,23)(22,24);(1,2,5,8)(3,4,7,6)(9,11,14,13)(10,12,15,16)(17,19)(18,22)(20,23)(21,24);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(18,23)(20,22);(9,11,14,13)(10,12,15,16)(17,20,21,22)(18,19,23,24);(1,3)(2,4)(5,7)(6,8)(9,13,14,11)(10,12,15,16)(17,19)(18,22)(20,23)(21,24);(1,4)(2,7)(3,8)(5,6)(9,12)(10,13)(11,15)(14,16)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=h,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7762698);