Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 | #37780 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2155 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)(13,14,18,22)(15,23,26,20)(16,27,28,25)(17,21,19,24);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,26)(14,20,22,23)(16,21,28,24)(17,25,19,27);(2,4)(13,16,17,23,18,28,19,20)(14,15,24,25,22,26,21,27);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,17,18,19)(14,21,22,24)(15,25,26,27)(16,20,28,23);(2,4)(15,26)(16,28)(20,23)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,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, 6491451);