Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 |
| prime radical | C2×#3558 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14,17,21,19,24,18,23)(15,22,25,28,16,27,26,20);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,16)(14,20,24,28)(17,26,18,25)(21,27,23,22);(6,9)(7,11)(14,21)(15,16)(17,18)(20,22)(23,24)(27,28);(5,7,8,11)(6,10,9,12)(13,16,19,15)(14,22,24,27)(17,26,18,25)(20,21,28,23);(6,9)(10,12)(15,16)(20,28)(22,27)(25,26)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420576);