Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
2437 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)(5,6,8,10)(7,11,12,9)(14,16)(15,19)(21,22,25,28)(23,24,27,26);(5,6,8,10)(7,9,12,11)(14,16)(18,20)(21,23)(22,26)(24,28)(25,27);(5,7,8,12)(6,9,10,11)(13,14,17,16)(15,20,19,18)(21,24,25,26)(22,27,28,23);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,19)(14,18,16,20)(21,23)(22,24)(25,27)(26,28);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,20,19,18)(21,24)(22,27)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=fg,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4064823);