Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,15)(14,19)(18,22)(20,24)(21,23);(10,15)(13,22)(16,24)(17,23)(18,19)(20,21);(2,5)(6,8)(9,11,14,12)(10,16,19,17)(13,21,22,20)(15,23,18,24);(9,12,14,11)(10,17,19,16)(13,20,22,21)(15,24,18,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,f^c=fi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279982, #72280013, #72280051, … |
| maximal normal subgroup of | #72279982, #72280013, #72280051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361029);