Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393925 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,22,18,23,28,20)(15,19,24,27,21,16,25,26);(6,9)(10,12)(13,15)(14,19)(16,23)(17,24)(18,21)(20,26)(22,27)(25,28);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(15,24)(17,28)(21,25)(22,23)(26,27);(13,16)(14,21)(15,23)(17,27)(18,19)(20,25)(22,24)(26,28)Polycyclic presentation:
2^9|a^2=f,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303005);