Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5095 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)(9,10,14,15)(11,13,12,16)(18,22)(23,24);(6,8)(10,15)(11,12)(17,18)(19,23)(20,24)(21,22);(5,6)(7,8)(9,11,14,12)(10,16,15,13)(17,19,21,20)(18,23,22,24);(1,2)(3,4)(9,12,14,11)(10,13,15,16)(17,20,21,19)(18,24,22,23);(9,13)(10,11)(12,15)(14,16)(17,18)(19,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72326339, #72326341, #72326376, … |
| maximal normal subgroup of | #72326339, #72326341, #72326376, … |
| maximal quotient of | #72330450, #72332482, #72333307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752511);