Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3063 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)(13,14,20,26,19,16,18,22)(15,23,28,21,24,27,25,17);(2,4)(6,10)(8,11)(13,15,19,24)(14,21,16,17)(18,28,20,25)(22,27,26,23);(5,6)(7,8)(9,10)(11,12)(13,16,19,14)(15,21,24,17)(18,22,20,26)(23,25,27,28);(5,7,9,12)(6,11,10,8)(13,17,19,21)(14,15,16,24)(18,27,20,23)(22,25,26,28);(5,8,9,11)(6,7,10,12)(15,24)(17,21)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524795);