Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1602 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)(5,6,8,9)(7,10,11,12)(13,14,18,22)(15,20,16,25)(17,23,24,21)(19,27,28,26);(6,9)(7,11)(13,15,24,27,18,16,17,26)(14,19,23,25,22,28,21,20);(6,9)(10,12)(13,16,18,15)(14,20,22,25)(17,27,24,26)(19,21,28,23);(2,4)(5,7,8,11)(6,10,9,12)(13,14)(15,25)(16,20)(17,23)(18,22)(19,27)(21,24)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337175);