Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
4674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,23,24,20);(6,8)(10,13)(14,15);(5,6)(7,8)(18,22)(20,23);(1,2)(3,4)(9,11,12,16)(10,14,13,15)(17,19)(18,20)(21,24)(22,23);(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=g,d^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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213568);