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 | |
6131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,6,7,8)(9,10,14,13)(11,15,12,16)(18,21)(22,24);(9,11,14,12)(10,15,13,16)(17,18)(19,22)(20,21)(23,24);(2,4)(6,8)(10,13)(11,12);(3,7)(6,8)(10,13)(15,16)(17,19,20,23)(18,22,21,24);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,16,13,15);(1,4,5,2)(3,8,7,6)(9,13,14,10)(11,16,12,15)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72258018, #72258911, #72262935, … |
| maximal normal subgroup of | #72258018, #72258911, #72262935, … |
| maximal quotient of | #72258897, #72258899 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7975429);