Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2404 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(6,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,24)(20,22)(21,23);(9,11)(10,13)(12,16)(14,15)(17,19,21,20)(18,22,23,24);(5,6)(7,8)(17,18)(19,24)(20,22)(21,23);(1,3,4,2)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,24)(21,23);(9,10)(11,13)(12,15)(14,16)(17,20,21,19)(18,22,23,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=i,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72232459, #72232479, #72232555, … |
| maximal normal subgroup of | #72232459, #72232479, #72232555, … |
| maximal quotient of | #72328804, #72330764, #72332949 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 425896);