Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34743 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,18,13,12,17,21)(11,15,20,22,19,16,23,24);(1,2,3,4)(9,11)(10,15)(12,16)(13,19)(14,20)(17,23)(18,22)(21,24);(6,8)(9,11)(10,16)(12,15)(13,19)(14,20)(17,23)(18,24)(21,22);(9,12,14,21,13,10,17,18)(11,15,20,22,19,16,23,24);(11,19)(15,16)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,d^b=di,e^b=ei,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239254, #72239271, #72250364, … |
| maximal normal subgroup of | #72239254, #72239271, #72250364, … |
| maximal quotient of | #72239336 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279971);