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 | |
11120 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)(3,4)(6,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24);(5,6)(7,8)(10,14)(15,17)(16,20)(22,23);(2,4)(9,10,13,14)(11,17,19,15)(12,16,21,20)(18,23,24,22);(1,2)(3,4)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)Polycyclic presentation:
2^9|c^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72316120, #72316203 |
| maximal normal subgroup of | #72316120, #72316203 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581320);