Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5670 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)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,24,23,20);(2,4)(6,8)(9,11,14,16)(10,15,12,13)(17,18)(19,24)(20,23)(21,22);(6,8)(10,12)(11,16)(18,22)(19,23);(1,2)(3,4)(5,6)(7,8)(9,12,14,10)(11,13,16,15)(17,19,21,23)(18,20,22,24);(9,13)(10,11)(12,16)(14,15)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72326415, #72326418, #72326446 |
| maximal normal subgroup of | #72326415, #72326418, #72326446 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836849);