Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C23×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,24)(20,22)(21,23);(2,4)(10,15)(11,19)(12,14)(16,23)(20,21)(22,24);(6,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,24)(18,23);(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,24)(18,23);(9,12,13,14)(10,17,15,18)(11,19,20,21)(16,24,22,23)Polycyclic presentation:
2^9|a^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282076, #72282085, #72282379, … |
| maximal normal subgroup of | #72282076, #72282085, #72282379, … |
| maximal quotient of | #72307655, #72311625, #72311893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344881);