Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| 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 | #24391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,17,19)(15,22,25,28)(16,20,27,26)(18,24,23,21);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,19)(16,27)(21,24)(22,28);(2,4)(13,15,23,20,17,25,18,26)(14,16,24,28,19,27,21,22);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,16,17,27)(14,20,19,26)(15,24,25,21)(18,22,23,28);(2,4)(15,25)(16,27)(20,26)(22,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491461);