Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | |
3419 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)(12,14)(13,18)(17,24)(19,21)(20,23);(1,2)(3,4)(5,6)(7,8)(9,10,15,18)(11,19,22,17)(12,16,14,13)(20,24,23,21);(6,8)(9,11,15,22)(10,17,18,19)(12,23,14,20)(13,21,16,24);(2,4)(6,8)(9,12,15,14)(10,16,18,13)(11,20,22,23)(17,21,19,24);(9,13)(10,12)(11,21)(14,18)(15,16)(17,23)(19,20)(22,24)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,d^c=di,f^e=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282065, #72282170, #72282179, … |
| maximal normal subgroup of | #72282065, #72282170, #72282179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421324);