Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4429 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)(6,8)(9,10,13,14)(11,15,16,12)(17,18,22,21)(19,24,23,20);(2,4)(10,14)(12,15);(1,2)(3,4)(17,19)(18,20)(21,24)(22,23);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,24)(18,23,21,19);(1,2)(3,4)(9,12)(10,11)(13,15)(14,16)(17,21,22,18)(19,24,23,20)Polycyclic presentation:
2^9|a^2=fgh,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232074, #72232109, #72232141, … |
| maximal normal subgroup of | #72232074, #72232109, #72232141, … |
| maximal quotient of | C2×#648318, #72329016, #72329855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 555514);