Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23386 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2267 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,23)(16,26)(17,27)(18,22)(19,20)(21,24)(25,28);(1,2)(3,4)(6,9)(7,12)(13,15,18,19)(14,20,22,23)(16,27,28,24)(17,26,21,25);(2,4)(13,16)(14,21)(15,24)(17,22)(18,28)(19,27)(20,26)(23,25);(5,7,8,12)(6,10,9,11)(13,17,18,21)(14,16,22,28)(15,25,19,26)(20,24,23,27);(14,22)(16,28)(20,23)(24,27)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,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, 7409530);