Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(10,12)(15,16)(17,18)(19,21)(20,23)(22,24);(2,3)(6,8)(9,10,14,12)(11,16,13,15)(17,19,20,24)(18,21,23,22);(1,2,4,3)(5,6)(7,8)(9,11,14,13)(10,15,12,16)(17,19)(18,22)(20,24)(21,23);(9,12,14,10)(11,16,13,15);(1,3,4,2)(9,13,14,11)(10,16,12,15)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=f,d^2=i,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4214760);