Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232313 |
| 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×#3192 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1812 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)(8,10,12,11)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,27)(26,28);(6,10)(7,11)(8,12)(13,15)(14,18)(16,21)(17,22)(19,26)(20,25)(23,28)(24,27);(2,4)(5,7)(6,9)(8,10)(11,12)(13,16)(14,19)(15,21)(17,23)(18,24)(20,27)(22,28)(25,26);(15,22)(18,25)(21,28)(24,26)Polycyclic presentation:
2^9|a^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=ei,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139510);