Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #265271 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #24140 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1154 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,15)(13,22)(14,18)(16,20)(17,23)(21,24);(1,3)(2,5)(4,7)(6,8)(9,11,13,12,14,20,23,21)(10,15,17,19,18,24,22,16);(5,8)(11,20)(12,21)(15,24)(16,19);(9,12,13,20,14,21,23,11)(10,16,17,15,18,19,22,24)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=dh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339524, #72339525, #72339526, … |
| maximal normal subgroup of | #72339524, #72339525, #72339526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 172467);