Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator, socle | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #52117 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2207 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)(6,9)(10,12)(13,14,19,22,18,21,28,25)(15,23,24,27,16,20,26,17);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,18)(17,22)(19,24)(21,23)(25,27)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,20,21,23)(17,25,27,22)(19,26,28,24);(6,9)(10,12)(14,21)(15,16)(22,25)(24,26);(13,17,19,23,18,27,28,20)(14,15,22,24,21,16,25,26)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,d^b=df,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72343080, #72343198 |
| maximal normal subgroup of | #72343080, #72343198 |
| maximal quotient of | #72343123, #72343868, #72343870 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291533);