Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 2097152 |
| 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×#2408 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4408 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)(6,8)(9,10,14,13)(11,12,16,15)(18,19)(20,24);(1,2,3,4)(9,11,14,16)(10,15,13,12)(18,19)(22,23);(1,2,3,4)(10,13)(11,16)(17,18,21,19)(20,22,24,23);(1,2,3,4)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,18)(20,23,24,22);(1,2)(3,4)(9,13,14,10)(11,15,16,12)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=fh,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72263083, #72263101, #72263102, … |
| maximal normal subgroup of | #72263083, #72263101, #72263102, … |
| maximal quotient of | #72298221, #72298316, #72333027 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3781584);