Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393999 |
| 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×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,20)(17,21)(18,22)(19,23)(25,26)(27,28);(5,7,8,11)(6,9,10,12)(13,15)(14,16)(17,20)(18,28)(19,25)(21,26)(22,24)(23,27);(1,2,3,4)(13,16,19,27)(14,20,23,28)(15,21,25,22)(17,26,18,24);(6,10)(9,12)(13,17,19,18)(14,21,23,22)(15,20,25,28)(16,26,27,24)Polycyclic presentation:
2^9|b^2=h,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313414);