Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2420 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)(6,9)(7,11)(14,16)(15,19)(21,22,26,25)(23,24,28,27);(5,6,8,9)(7,12,11,10)(15,19)(18,20)(21,23,26,28)(22,27,25,24);(6,9)(10,12)(13,14,17,16)(15,18,19,20)(21,23,26,28)(22,27,25,24);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,17,19)(14,18,16,20)(21,24)(22,23)(25,28)(26,27);(5,6)(7,12)(8,9)(10,11)(13,16,17,14)(15,20,19,18)(21,25,26,22)(23,27,28,24)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=g,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3767107);