Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 | C2×#3204 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,22,23,27)(16,21,18,25)(19,26,28,24);(1,2)(3,4)(6,9)(7,11)(13,14)(15,19)(16,25)(17,20)(18,21)(22,24)(23,28)(26,27);(2,4)(13,15,16,26,17,23,18,24)(14,19,25,27,20,28,21,22);(6,9)(10,12)(14,20)(15,23)(21,25)(24,26);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,21,20,25)(15,24,23,26)(19,27,28,22)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592527);