Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | #39954 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,12,8,10);(5,6)(7,10)(8,12)(9,11)(13,14,18,22)(15,20,17,19)(16,24,26,27)(21,25,28,23);(1,2,3,4)(14,19)(15,17)(20,22)(21,24)(23,25)(27,28);(5,7,9,8)(6,10,11,12)(13,15,18,17)(14,20,22,19)(16,25,26,23)(21,24,28,27);(5,8,9,7)(6,10,11,12)(13,16,18,26)(14,21,22,28)(15,23,17,25)(19,24,20,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594243);