Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,17,14,18,12,15)(11,16,21,24,19,23,20,22);(2,4)(10,15,18,17)(16,22,23,24);(5,6,7,8)(11,19)(16,23)(20,21)(22,24);(9,11,13,21,14,19,12,20)(10,16,17,24,18,23,15,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=f,e^2=i,f^2=i,b^a=be,e^a=ei,g^a=gi,d^c=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72214887, #72224689, #72224693, … |
| maximal normal subgroup of | #72214887, #72224689, #72224693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260407);