Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,12)(9,14)(10,15)(11,16)(13,18)(19,20)(21,22,25,29)(23,26,30,35)(24,32,34,28)(27,33,36,31);(1,2)(3,4)(5,7)(6,12)(8,9)(10,18)(11,19)(13,16)(14,20)(15,17)(21,23)(22,26)(24,31)(25,30)(27,32)(28,36)(29,35)(33,34);(2,4)(5,8,10,19,11,18,9,7)(6,13,14,20,16,12,15,17)(21,23)(22,27)(24,33)(25,30)(26,32)(28,35)(29,36)(31,34);(5,9,11,10)(6,14,16,15)(7,18,19,8)(12,17,13,20)(23,30)(26,35)(27,36)(31,33)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,d^c=di,f^c=fhi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147708);