Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,10)(11,12)(13,14,18,22,20,23,19,24)(15,21,25,28,26,27,17,16);(1,3,4,2)(5,6)(7,11)(8,12)(9,10)(13,15)(14,16)(17,18)(19,25)(20,26)(21,24)(22,27)(23,28);(2,3)(5,7,9,8)(6,11,10,12)(13,16,19,27,20,28,18,21)(14,15,24,17,23,26,22,25);(5,8,9,7)(6,11,10,12)(13,17,19,26,20,25,18,15)(14,21,24,16,23,27,22,28)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=ei,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340485);