Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(10,13)(14,16)(19,23)(22,24);(5,6,7,8)(18,21)(22,24);(1,2)(3,4)(17,18)(19,22)(20,21)(23,24);(9,10)(11,14)(12,13)(15,16);(9,11,12,15)(10,14,13,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|b^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72297682, #72297688, #72297691, … |
| maximal normal subgroup of | #72297682, #72297688, #72297691, … |
| maximal quotient of | #72298559, #72323571, C2×#4308704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215900);