Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2669 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(9,10,13,11)(12,14,16,15)(17,18,20,23)(19,22,24,21);(1,2)(3,4)(10,11)(14,15)(17,19)(18,21)(20,24)(22,23);(2,4)(12,16)(14,15)(19,24)(21,22);(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,23)(21,24);(1,2)(3,4)(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72327164, #72327247, #72327274 |
| maximal normal subgroup of | #72327164, #72327247, #72327274 |
| maximal quotient of | #72328814 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1991394);