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×#3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3019 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)(13,14,18,22)(15,23,19,27)(16,26,24,25)(17,20,28,21);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,19)(14,16,22,24)(17,26,28,25)(20,23,21,27);(6,9)(10,12)(13,16,18,24)(14,19,22,15)(17,27,28,23)(20,26,21,25);(5,7)(6,10)(8,11)(9,12)(13,14)(15,24)(16,19)(17,21)(18,22)(20,28)(23,26)(25,27);(7,11)(10,12)(13,17,18,28)(14,20,22,21)(15,25,19,26)(16,27,24,23)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,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, 6449486);