Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3016 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)(7,11)(10,12)(13,14)(15,23)(16,24)(17,21)(18,20)(19,22)(25,27)(26,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,16,22,27)(17,24,18,25)(20,23,21,28);(6,9)(10,12)(13,16)(14,15)(17,23)(18,28)(19,27)(20,24)(21,25)(22,26);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,20,22,21)(15,24,26,25)(16,23,27,28);(13,17,19,18)(14,21,22,20)(15,25,26,24)(16,23,27,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342648);