Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233431 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3205 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,17,22)(15,23,26,20)(16,19,18,21)(24,28,25,27);(2,4)(5,6)(7,9)(8,11)(10,12)(14,19)(15,24)(16,18)(21,22)(25,26)(27,28);(6,9)(7,12)(10,11)(13,15)(14,20)(16,24)(17,26)(18,25)(19,27)(21,28)(22,23);(13,16,17,18)(14,21,22,19)(15,24,26,25)(20,28,23,27)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140630);