Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,18)(12,14)(13,15)(16,23)(17,21)(19,22)(20,24);(5,6)(7,8)(10,14)(11,19)(12,21)(15,17)(18,24)(20,23);(2,4)(6,8)(10,15)(14,17)(16,22)(18,24);(2,4)(9,11,12,19,13,20,21,23)(10,16,17,24,15,22,14,18);(11,20)(16,22)(18,24)(19,23)Polycyclic presentation:
2^9|d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,c^b=ch,d^b=df,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72312918, #72312930, #72313009 |
| maximal normal subgroup of | #72312918, #72312930, #72313009 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409329);