Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,8)(7,11,12,10)(13,14,17,18)(15,19,20,16)(21,22,25,27)(23,24,28,26);(2,4)(14,18)(16,19)(21,23,25,28)(22,26,27,24);(1,2)(3,4)(21,24)(22,23)(25,26)(27,28);(5,7,9,12)(6,10,8,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28);(5,8,9,6)(7,11,12,10)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=fi,e^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232618, #72232636, #72233131, … |
| maximal normal subgroup of | #72232618, #72232636, #72233131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 593385);