Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,12,14)(11,15,16,13)(17,18)(19,21)(20,23)(22,24);(6,8)(11,16)(13,15)(18,19)(20,24);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,19)(20,23,24,22);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,22,24,23);(9,11)(10,13)(12,16)(14,15)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|a^2=fg,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266747, #72266998 |
| maximal normal subgroup of | #72266747, #72266998 |
| maximal quotient of | #72267002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3941071);