Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,20,16)(21,22)(23,24)(25,27)(26,28);(6,9)(10,12)(13,14)(15,19)(16,20)(17,18)(21,23,26,27)(22,25,28,24);(7,11)(10,12)(14,18)(16,19)(21,24,26,25)(22,23,28,27);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,15)(14,19)(16,18)(17,20)(21,25,26,24)(22,27,28,23);(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,20,18,15)(21,22)(23,25)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3271641);