Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3036 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,24)(21,23);(6,7)(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22);(1,2,3,4)(5,6,8,7)(10,15)(11,18)(16,20)(19,21);(5,7,8,6)(11,18)(14,23)(17,24)(19,21);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|c^2=gh,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316431, #72316450, #72342363, … |
| maximal normal subgroup of | #72316431, #72316450, #72342363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372678);