Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232291 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3100 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,7)(8,11,10,12)(13,14,17,20)(15,22,24,23)(16,27,18,21)(19,26,28,25);(6,11)(7,12)(8,10)(13,15)(14,19)(16,26)(17,24)(18,25)(20,28)(21,23)(22,27);(2,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,23)(16,21)(17,20)(18,27)(19,25)(22,24)(26,28);(5,8,9,10)(6,12,7,11)(14,20)(15,24)(21,27)(25,26)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,d^c=dh,e^c=ei,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139502);