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 | #59057 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1994 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)(15,23,24,28)(16,21,17,20)(19,25,27,26);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(15,24)(19,27)(23,28)(25,26);(5,7)(6,11)(8,10)(9,12)(13,15,17,26,18,24,16,25)(14,19,21,23,22,27,20,28);(2,4)(6,10)(7,12)(15,24)(19,27)(23,28)(25,26);(5,8,9,11)(6,7,10,12)(13,16,18,17)(14,20,22,21)(15,25,24,26)(19,28,27,23)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=g,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, 7092416);