Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(13,14,15,20,18,22,19,27)(16,24,25,28,17,26,23,21);(1,2,4,3)(5,6,8,10)(7,11,12,9)(13,15,18,19)(14,20,22,27)(16,25,17,23)(21,24,28,26);(5,7,8,12)(6,9,10,11)(13,16,18,17)(14,21,22,28)(15,23,19,25)(20,26,27,24);(1,3,4,2)(13,17,18,16)(14,21,22,28)(15,23,19,25)(20,24,27,26)Polycyclic presentation:
2^9|a^2=f,b^2=eg,c^2=g,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^c=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310894);