Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| 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 | C2×#3050 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2286 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)(13,14,18,20)(15,24,22,27)(16,26,21,25)(17,23,19,28);(1,2)(3,4)(6,9)(7,12)(14,20)(16,21)(23,28)(24,27);(2,4)(13,15,18,22)(14,21,20,16)(17,26,19,25)(23,27,28,24);(5,7,8,12)(6,10,9,11)(13,16,18,21)(14,22,20,15)(17,27,19,24)(23,26,28,25);(13,17,18,19)(14,23,20,28)(15,25,22,26)(16,27,21,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409520);