Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| 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 | C2×#3319 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1930 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)(5,6)(7,9)(8,10)(11,12)(13,14,17,21,18,20,25,16)(15,22,24,27,23,28,26,19);(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,22)(17,26)(18,23)(20,27)(21,28)(24,25);(2,4)(5,7,8,12)(6,9,10,11)(14,20)(15,23)(16,21)(24,26);(7,12)(9,11)(13,16,25,20,18,21,17,14)(15,19,26,28,23,27,24,22);(6,10)(9,11)(15,23)(19,27)(22,28)(24,26)Polycyclic presentation:
2^9|a^2=fgi,c^2=hi,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972033);