Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,10,8,12)(13,14,18,22,20,23,19,16)(15,24,17,21,27,28,26,25);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,20,27)(14,21,23,25)(16,28,22,24)(17,18,26,19);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,14,20,22,19,23)(15,25,17,24,27,21,26,28);(2,4)(13,17,19,15,20,26,18,27)(14,21,16,24,23,25,22,28);(13,18,20,19)(14,22,23,16)(15,26,27,17)(21,24,25,28)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355344);