Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 | #31068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1919 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)(7,8)(9,11)(10,12)(13,14,17,21,18,22,26,27)(15,19,23,20,16,24,25,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,18)(17,23)(20,21)(22,24)(25,26)(27,28);(2,4)(13,16,18,15)(14,20,22,28)(17,23,26,25)(19,27,24,21);(6,11)(7,12)(14,21)(17,26)(19,20)(22,27)(23,25)(24,28);(5,8,9,10)(6,7,11,12)(14,22)(15,16)(21,27)(23,25)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,e^b=ei,e^c=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, 7411223);