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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2029 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)(13,14,20,23,19,17,18,24)(15,25,22,28,26,21,27,16);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,21)(16,23)(17,25)(18,27)(19,26)(20,22)(24,28);(2,4)(5,8,9,12)(6,10,7,11)(13,16,19,28)(14,22,17,27)(15,24,26,23)(18,25,20,21);(6,7)(10,11)(13,17)(14,19)(15,21)(16,27)(18,24)(20,23)(22,28)(25,26);(13,18,19,20)(14,23,17,24)(15,22,26,27)(16,21,28,25)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7410854);