Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #759288 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7314 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)(6,8)(9,10,13,16)(11,15,12,14)(18,20)(19,23);(9,11,13,12)(10,14,16,15)(17,18,21,20)(19,22,23,24);(5,6)(7,8)(11,12)(14,15)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16);(9,12,13,11)(10,15,16,14)(17,20,21,18)(19,22,23,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237102, #72237107, #72237111, … |
| maximal normal subgroup of | #72237102, #72237107, #72237111, … |
| maximal quotient of | #72298308, #72298333, #72298468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 666660);