Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(17,18)(19,23)(20,24)(21,22);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,20,21)(18,22,24,23);(10,13)(11,15)(18,22)(19,21)(23,24);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,19,20,21)(18,23,24,22);(10,13)(11,15)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ehi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72305296, #72305303, #72305306 |
| maximal normal subgroup of | #72305296, #72305303, #72305306 |
| maximal quotient of | #72306007, #72306034, #72316006 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6507920);