Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| 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×#2424 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,8)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,23,24,20);(1,2,4,3)(9,10)(11,13)(12,14)(15,16)(19,24)(20,23);(2,3)(11,16)(13,15)(17,19)(18,20)(21,24)(22,23);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,23)(22,24);(1,3,4,2)(9,11)(10,13)(12,16)(14,15)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72331404, #72331414, #72331423 |
| maximal normal subgroup of | #72331404, #72331414, #72331423 |
| maximal quotient of | #72333286 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 908906);