Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| 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 | #16250 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,9)(10,11)(14,17)(15,19)(21,22,24,26)(23,27,28,25);(5,6,10,11)(7,12,8,9)(15,19)(18,20)(21,22)(23,25)(24,26)(27,28);(5,7,10,8)(6,9,11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,26)(25,28);(1,2,3,4)(5,8,10,7)(6,9,11,12)(13,15,16,19)(14,18,17,20);(5,9)(6,7)(8,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=h,d^2=hi,e^2=fh,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, 2236266);