Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35222 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(11,18)(12,13)(15,22)(16,20)(19,21)(23,24);(1,2,4,3)(5,6)(7,8)(9,10,14,17)(11,15,19,24)(12,20,13,16)(18,22,21,23);(1,3,4,2)(6,8)(9,11)(10,15)(12,18)(13,21)(14,19)(16,23)(17,24)(20,22);(6,8);(9,12,14,13)(10,16,17,20)(11,18,19,21)(15,23,24,22)Polycyclic presentation:
2^9|b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72313286, #72313301, #72313314 |
| maximal normal subgroup of | #72313286, #72313301, #72313314 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345994);