Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
4670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(18,19)(22,23);(5,6)(7,8)(10,14)(12,15)(17,18,21,19)(20,23,24,22);(2,4)(6,8)(10,14)(12,15);(9,11,13,16)(10,15,14,12)(17,19,21,18)(20,23,24,22);(1,2)(3,4)(5,6)(7,8)(9,12,13,15)(10,16,14,11)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|b^2=i,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
| maximal subgroup of | #72264366, #72264446, #72264460, … |
| maximal normal subgroup of | #72264366, #72264446, #72264460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213733);