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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1931 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)(6,10)(8,11)(13,14,18,22)(15,19,17,21)(16,24,26,27)(20,23,28,25);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(13,15,18,17)(14,19,22,21)(16,25,26,23)(20,27,28,24);(5,7)(6,11)(8,10)(9,12)(13,16,15,23,18,26,17,25)(14,20,21,27,22,28,19,24);(2,4)(6,10)(7,12)(16,26)(20,28)(23,25)(24,27);(5,8,9,11)(6,7,10,12)(13,15,18,17)(14,21,22,19)(16,23,26,25)(20,27,28,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092418);