Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3304 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)(10,16)(11,19)(12,14)(13,18)(17,21)(23,24);(1,2)(3,4)(5,6)(7,8)(9,10,15,18)(11,17,19,24)(12,16,14,13)(20,21,22,23);(6,8)(9,11,15,19)(10,17,18,24)(12,22,14,20)(13,21,16,23);(2,4)(6,8)(9,12,15,14)(10,16,18,13)(11,20,19,22)(17,21,24,23);(9,13)(10,12)(11,21)(14,18)(15,16)(17,22)(19,23)(20,24)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,d^c=di,f^e=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282085, #72282214, #72282753, … |
| maximal normal subgroup of | #72282085, #72282214, #72282753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421502);