Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,18)(12,17)(13,21)(14,16)(15,19)(20,24)(22,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(2,3)(7,8)(10,16)(15,23)(17,21)(18,24);(10,16)(15,23)(17,21)(18,24);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,17,16,21)(11,19,20,22)(15,24,23,18)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dhi,f^a=fh,c^b=ci,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239236, #72239272, #72239292, … |
| maximal normal subgroup of | #72239236, #72239272, #72239292, … |
| maximal quotient of | #72239262 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250208);