Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
11352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(5,6,7,8)(9,10)(11,16)(12,20)(13,15)(14,18)(17,21)(19,22)(23,24);(5,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,16)(17,22)(20,23)(21,24);(2,3)(10,15)(11,18)(17,20)(19,24);(1,3,4,2)(9,11)(10,16)(12,19)(13,18)(14,15)(17,23)(20,22)(21,24);(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,21)(22,24);(9,12,13,21)(10,17,15,20)(11,19,18,24)(14,22,16,23)Polycyclic presentation:
2^9|a^2=gh,d^2=hi,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72234472, #72234475, #72234488, … |
| maximal normal subgroup of | #72234472, #72234475, #72234488, … |
| maximal quotient of | #72257053, #72257078, #72257118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7588646);