Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| 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×#2539 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3638 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)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,24,21,23);(1,2,3,4)(10,13)(14,15)(17,19)(18,23)(20,24)(21,22);(5,6,8,7)(17,18,22,20)(19,24,21,23);(1,2,3,4)(5,6,8,7)(9,11,12,16)(10,14,13,15)(17,20,22,18)(19,24,21,23);(1,2)(3,4)(5,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280903, #72281105, #72281261 |
| maximal normal subgroup of | #72280903, #72281105, #72281261 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1353640);