Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 | #39958 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1933 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,16,21,17,18,26,20)(15,22,25,27,23,28,24,19);(2,3)(5,6)(7,11)(8,10)(9,12)(14,18)(15,23)(20,21)(24,25);(5,7,8,12)(6,9,10,11)(13,15,17,23)(14,19,18,27)(16,24,26,25)(20,22,21,28);(1,3,4,2)(7,12)(9,11)(14,18)(15,23)(20,21)(24,25);(6,10)(9,11)(13,16,17,26)(14,20,18,21)(15,24,23,25)(19,22,27,28)Polycyclic presentation:
2^9|a^2=fgi,c^2=h,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6971838);