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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,10)(8,11)(13,14,18,16)(15,21,22,28)(17,25,27,20)(19,26,24,23);(5,6,9,10)(7,8,12,11)(13,15)(14,19)(16,24)(17,26)(18,22)(20,28)(21,25)(23,27);(2,4)(5,7)(6,11)(8,10)(9,12)(15,22)(19,24)(21,28)(23,26);(6,10)(7,12)(13,16,18,14)(15,21,22,28)(17,20,27,25)(19,26,24,23);(5,8,9,11)(6,7,10,12)(15,22)(19,24)(21,28)(23,26)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089272);