Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3396 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)(6,7)(9,10,13,12)(11,17,20,15)(14,16,23,22)(18,21,24,19);(5,6,8,7)(9,11)(10,15)(12,17)(13,20)(14,19)(16,24)(18,22)(21,23);(2,4)(5,7,8,6)(9,12,13,10)(11,18,20,24)(14,16,23,22)(15,19,17,21);(11,19)(14,23)(15,24)(16,22)(17,18)(20,21);(10,12)(11,20)(16,22)(19,21)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72312850, #72313036, #72313045, … |
| maximal normal subgroup of | #72312850, #72313036, #72313045, … |
| maximal quotient of | #72315786 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345917);