Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 | #59145 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1910 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)(15,22)(16,20)(17,26)(18,21)(19,23)(24,27)(25,28);(1,2)(3,4)(5,6,9,10)(7,8,12,11);(5,7)(6,11)(8,10)(9,12)(13,15,17,23,18,24,16,25)(14,19,20,22,21,28,26,27);(2,4)(6,10)(7,12)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,28,22);(5,8,9,11)(6,7,10,12)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,28,22)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092680);