Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,10)(7,11)(14,17)(15,19)(21,22,26,28)(23,25,24,27);(5,6,9,10)(7,12,11,8)(14,17)(18,20)(21,22)(23,25)(24,27)(26,28);(6,10)(8,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,24)(22,27,28,25);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,24,26,23)(22,27,28,25);(5,8)(6,7)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,25,26,27)(22,23,28,24)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=fgh,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2047173);