Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,25)(17,20)(18,22)(19,24)(21,26)(27,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,17,24)(14,19,20,23)(16,18,26,27)(21,22,25,28);(6,10)(7,11)(14,20)(15,24)(16,26)(22,28);(6,10)(9,12)(13,16,18,24,17,26,27,15)(14,21,22,23,20,25,28,19);(1,3,4,2)(14,20)(19,23)(21,25)(22,28)Polycyclic presentation:
2^9|b^2=f,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518681);