Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(10,15,18,17)(16,23,24,19);(1,2,4,3)(5,6)(7,8)(9,10)(11,19)(12,17)(13,15)(14,18)(16,22)(20,23)(21,24);(1,3,4,2)(6,8)(9,11,12,21,14,20,13,22)(10,16,15,23,18,24,17,19);(11,20)(16,24)(19,23)(21,22)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239933 |
| maximal normal subgroup of | #72239933 |
| maximal quotient of | #72239947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265505);