Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5008 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)(3,4)(6,8)(9,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,24)(22,23);(2,4)(9,11)(10,15)(12,21)(13,20)(14,22)(16,24)(17,23)(18,19);(5,6,7,8)(11,20)(12,13)(15,19)(17,18)(21,22)(23,24);(11,21)(12,13)(15,23)(17,18)(19,24)(20,22);(10,16)(15,24)(17,18)(19,23)Polycyclic presentation:
2^9|c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237163, #72249392, #72249414, … |
| maximal normal subgroup of | #72237163, #72249392, #72249414, … |
| maximal quotient of | #72237132, C2×#6442649, #72308477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349845);