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 | |
5997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,7)(9,10)(11,14)(12,15)(13,16)(18,21)(22,24);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,14,16,15)(17,18,20,21)(19,22,23,24);(2,4)(6,7)(9,10,13,16)(11,14,12,15);(3,8)(6,7)(9,10)(11,14)(12,15)(13,16);(1,3,5,8)(2,6,4,7);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,16,14)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72297326, #72312017, #72312019, … |
| maximal normal subgroup of | #72297326, #72312017, #72312019, … |
| maximal quotient of | #72297399, #72297939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7590426);