Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(9,10)(11,19)(12,18)(13,17)(14,16)(15,21)(20,24)(22,23);(1,2)(3,4)(6,7)(9,11,12,22,14,20,13,21)(10,15,17,24,16,23,18,19);(5,6,8,7)(10,16)(11,20)(17,18)(21,22);(2,4)(5,7,8,6)(9,12,14,13)(10,17,16,18)(11,21,20,22)(15,19,23,24);(9,11,12,22,14,20,13,21)(10,15,17,24,16,23,18,19)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257084, #72257113, #72257202, … |
| maximal normal subgroup of | #72257084, #72257113, #72257202, … |
| maximal quotient of | #72264863, #72308594 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470467);