Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(14,18)(15,20);(6,7)(10,11)(21,22,24,26)(23,27,28,25);(5,6,9,7)(8,10,12,11)(13,14,17,18)(15,19,20,16)(21,23,24,28)(22,25,26,27);(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,18)(17,20);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,15,18,20)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fgh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428722);