Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #763332 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10)(11,15)(12,13)(14,16)(18,20)(19,23);(10,13)(14,15)(17,18,21,20)(19,24,23,22);(5,6)(7,8)(11,16)(14,15)(18,20)(22,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23);(9,11)(10,14)(12,16)(13,15)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^9|b^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219260, #72219267, #72219268, … |
| maximal normal subgroup of | #72219260, #72219267, #72219268, … |
| maximal quotient of | #72323513, #72323536, #72323722 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 672719);