Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394040 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,18)(12,22)(13,16)(14,17)(15,19)(20,23)(21,24);(9,11)(10,15)(12,21)(13,19)(14,20)(16,23)(17,24)(18,22);(11,19)(12,13)(15,23)(16,22)(18,24)(20,21);(1,3,4,8)(2,5,6,7)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72252474, #72252481, #72252490, … |
| maximal normal subgroup of | #72252474, #72252481, #72252490, … |
| maximal quotient of | C2×#414691 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321887);