Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | C2×#2428 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2954 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)(7,9,12,11)(14,16)(15,20)(22,24)(23,25);(5,6)(7,9)(8,10)(11,12)(13,14,17,16)(15,19,20,18)(21,22,26,24)(23,27,25,28);(5,6,8,10)(7,9,12,11)(13,15,17,20)(14,18,16,19)(21,23,26,25)(22,27,24,28);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,24,26,22)(23,27,25,28);(5,7)(6,9)(8,12)(10,11)(13,16,17,14)(15,19,20,18)(21,25,26,23)(22,28,24,27)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1316794);