Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3351 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)(10,15)(11,19)(12,13)(16,24)(17,18)(20,23);(2,4)(6,7)(9,10,12,17,14,18,13,15)(11,20,22,23,19,16,21,24);(5,6,8,7)(9,11,14,19)(10,16,18,20)(12,22,13,21)(15,23,17,24);(1,2,3,4)(5,7,8,6)(9,12,14,13)(10,17,18,15)(11,21,19,22)(16,23,20,24);(9,10,12,17,14,18,13,15)(11,16,22,24,19,20,21,23)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=hi,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,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72251200, #72256885, #72256935, … |
| maximal normal subgroup of | #72251200, #72256885, #72256935, … |
| maximal quotient of | C2×#7614034 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521230);