Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14,18,22,17,21,20,16)(15,23,26,27,25,19,28,24);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,25)(14,19,21,23)(16,27,22,24)(18,26,20,28);(2,4)(5,8,9,12)(6,10,7,11)(13,16)(14,20)(15,24)(17,22)(18,21)(19,26)(23,28)(25,27);(6,7)(10,11)(13,14,17,21)(15,23,25,19)(16,20,22,18)(24,28,27,26);(15,25)(19,23)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410892);