Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(10,14)(12,20)(15,21)(16,22)(17,24)(19,23);(5,6)(7,8)(9,10,12,14,13,16,20,22)(11,17,19,24,18,15,23,21);(2,4)(6,8)(9,11)(10,15)(12,19)(13,18)(14,21)(16,17)(20,23)(22,24);(5,6)(7,8)(10,16)(11,18)(14,22)(19,23);(6,8)Polycyclic presentation:
2^9|b^2=f,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ehi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257534, #72257677, #72263856, … |
| maximal normal subgroup of | #72257534, #72257677, #72263856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481526);