Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(9,10)(11,20)(12,18)(13,23)(14,24)(15,17)(16,21)(19,22);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,17,19)(12,21,14,22)(18,23,24,20);(6,8)(10,17)(16,19)(18,24)(20,23);(2,4)(6,8)(9,12,15,14)(10,18,17,24)(11,21,13,22)(16,23,19,20);(9,13,15,11)(10,19,17,16)(12,22,14,21)(18,20,24,23)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,f^e=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72281906, #72282076, #72282133, … |
| maximal normal subgroup of | #72281906, #72282076, #72282133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421214);