Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(9,10,15,19)(11,12,17,21)(13,22,20,16)(14,23,24,18);(1,2)(3,4)(6,7)(9,11,14,20,15,17,24,13)(10,16,18,12,19,22,23,21);(5,6,8,7)(9,12)(10,17)(11,19)(13,18)(14,22)(15,21)(16,24)(20,23);(2,4)(5,7,8,6)(11,17)(12,21)(13,20)(16,22);(9,13,14,11,15,20,24,17)(10,12,18,22,19,21,23,16)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283132, #72284003, #72284239 |
| maximal normal subgroup of | #72283132, #72284003, #72284239 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470444);