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 | |
3524 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)(11,18)(12,20)(13,24)(14,17)(15,21)(16,22)(19,23);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,21,19)(12,22,14,23)(17,18,20,24);(6,8)(9,12,15,14)(10,17,21,20)(11,22,13,23)(16,18,19,24);(2,4)(6,8)(9,11,15,13)(10,18,21,24)(12,23,14,22)(16,20,19,17);(9,13,15,11)(10,19,21,16)(12,23,14,22)(17,24,20,18)Polycyclic presentation:
2^9|b^2=i,c^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,d^c=di,f^e=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72281707, #72281759, #72282065, … |
| maximal normal subgroup of | #72281707, #72281759, #72282065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421254);