Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #39137 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1923 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)(5,6,10,9)(7,12,8,11)(13,14,18,21)(15,22,25,24)(16,23,20,17)(19,27,28,26);(5,7,10,8)(6,11,9,12)(13,15,23,27,18,25,17,26)(14,19,16,24,21,28,20,22);(2,4)(5,6,10,9)(7,12,8,11)(13,16,17,14,18,20,23,21)(15,24,26,19,25,22,27,28);(5,8,10,7)(6,11,9,12)(15,25)(19,28)(22,24)(26,27);(5,9,10,6)(7,12,8,11)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=f,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6474724);