Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| 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 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4542 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)(11,15)(12,13)(14,16)(17,18)(19,24)(20,23)(21,22);(1,2,4,3)(9,10,13,12)(11,15,16,14)(17,19,22,20)(18,23,21,24);(2,3)(10,12)(14,15)(17,19,22,20)(18,24,21,23);(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,20,22,19)(18,24,21,23);(1,3,4,2)(9,12,13,10)(11,15,16,14)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^9|b^2=f,c^2=fh,d^2=fgh,e^2=f,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 | #72263595, #72263606, #72263618, … |
| maximal normal subgroup of | #72263595, #72263606, #72263618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 903231);