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 | |
6449 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)(11,16)(12,14)(13,15)(17,18)(19,23)(20,22)(21,24);(1,2)(3,7)(4,5)(6,8)(9,11)(10,12)(13,14)(15,16)(17,19)(18,21)(20,24)(22,23);(2,5)(6,7)(11,14)(12,16)(18,22)(21,23);(1,3)(2,6)(4,8)(5,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,21)(20,22)(23,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,16)(12,14)(13,15)(17,19)(18,21)(20,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,12,13,16)(10,14,15,11)Polycyclic presentation:
2^9|f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72249410, #72249496, #72249512, … |
| maximal normal subgroup of | #72249410, #72249496, #72249512, … |
| maximal quotient of | #72263510, #72281919, #72300528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7660017);