Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,16,14,17,12,22)(11,18,20,23,21,24,19,15);(2,4)(9,11,14,21)(10,15)(12,19,13,20)(16,18)(17,23)(22,24);(6,8)(9,10)(11,15)(12,16)(13,22)(14,17)(18,19)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^b=fi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239546, #72239555, #72239564, … |
| maximal normal subgroup of | #72239546, #72239555, #72239564, … |
| maximal quotient of | #72239703, #72240292, #72249724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30810);