Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3223 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)(5,6,8,10)(7,9,12,11)(14,16)(15,20)(21,22,25,28)(23,26,27,24);(5,6)(7,9)(8,10)(11,12)(13,14,17,16)(15,19,20,18)(21,23,25,27)(22,26,28,24);(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,27)(25,28);(5,7,8,12)(6,9,10,11)(13,16,17,14)(15,19,20,18)(21,24,25,26)(22,27,28,23)Polycyclic presentation:
2^9|a^2=fg,b^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1172478);