Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| 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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5318 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)(9,10)(11,19)(12,15)(13,18)(14,17)(16,20)(21,24)(22,23);(5,6)(7,8)(10,15)(11,20)(12,13)(16,23)(17,18)(21,22);(2,4)(6,8)(9,11,12,22,14,21,13,20)(10,16,18,24,17,23,15,19);(11,21)(16,23)(19,24)(20,22);(10,17)(15,18)(16,23)(19,24)Polycyclic presentation:
2^9|c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=gi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249533, #72249569, #72249575, … |
| maximal normal subgroup of | #72249533, #72249569, #72249575, … |
| maximal quotient of | #72264892, #72308823, #72310591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339161);