Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1530 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)(6,9)(10,12)(13,14)(15,20)(16,26)(17,22)(18,23)(19,24)(21,27)(25,28);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,24)(17,25)(18,21)(19,26)(22,28)(23,27);(5,7)(6,10)(8,11)(9,12)(13,16,19,20)(14,21,24,25)(15,23,26,22)(17,27,18,28);(7,11)(10,12)(13,17,19,18)(14,22,24,23)(15,25,26,21)(16,27,20,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287097);