Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2631 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,19)(18,20)(21,22,26,24)(23,28,25,27);(6,9)(10,12)(13,15)(14,18)(16,19)(17,20)(22,24)(23,25);(5,7,8,11)(6,10,9,12)(13,14,17,19)(15,18,20,16)(21,23,26,25)(22,27,24,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,16)(17,19)(18,20)(21,24,26,22)(23,27,25,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25,26,23)(22,27,24,28)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=h,e^2=h,b^a=bf,c^a=cg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 463703);