Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3562 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(14,15)(17,18)(20,23)(21,22)(24,28)(25,26);(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,22)(15,18,21,17)(16,24,27,20)(23,26,28,25);(6,9)(10,12)(13,15,19,21)(14,17,22,18)(16,23,27,28)(20,26,24,25);(5,7)(6,10)(8,11)(9,12)(13,16,19,27)(14,20,22,24)(15,23,21,28)(17,26,18,25);(7,11)(10,12)(13,17,19,18)(14,15,22,21)(16,25,27,26)(20,28,24,23)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449948);