Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6)(7,8)(10,14)(15,16)(18,21)(22,23);(1,3,4,2)(9,10,13,14)(11,15,12,16)(17,18,19,21)(20,22,24,23);(2,3)(9,11,13,12)(10,15,14,16)(18,22)(20,24)(21,23);(6,8)(9,12,13,11)(10,15,14,16)(18,21)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=ei,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265394, #72265413, #72265467, … |
| maximal normal subgroup of | #72265394, #72265413, #72265467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329757);