Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394746 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#400 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,23,14,17,20,18)(11,19,12,15,21,24,22,16);(1,3,4,7)(2,5,6,8)(9,11,20,22,14,21,13,12)(10,15,23,24,17,16,18,19);(9,12,13,21,14,22,20,11)(10,16,18,24,17,15,23,19);(2,6)(5,8)(10,17)(15,16)(18,23)(19,24)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265212, #72265323, #72265332, … |
| maximal normal subgroup of | #72265212, #72265323, #72265332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 316490);