Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| 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 | #23422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,18,22,19,15,17,21)(16,20,26,28,24,23,25,27);(1,2)(3,4)(6,9)(10,12)(13,15,19,14)(16,23,24,20)(17,21,18,22)(25,27,26,28);(2,4)(13,15)(14,19)(16,23)(17,21)(18,22)(20,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,24)(21,27)(22,28);(16,24)(20,23)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339817);