Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2886 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)(10,15)(11,18)(12,13)(16,23)(17,21)(19,20);(5,6,7,8)(9,10,14,17)(11,16,19,23)(12,21,13,15)(18,22,20,24);(2,4)(9,11,13,18,14,19,12,20)(10,16,15,22,17,23,21,24);(11,19)(16,23)(18,20)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239049 |
| maximal normal subgroup of | #72239049 |
| maximal quotient of | #72239052, #72239081, #72239082 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261306);