Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#449 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,10,12,15)(7,13,14,16)(21,22,25,27,26,29,24,30)(23,31,34,28,32,35,33,36);(1,2,3,4)(5,6,8,15,9,12,17,10)(7,11,16,20,14,19,13,18)(22,27)(24,25)(28,31)(29,30)(33,34)(35,36);(5,7)(6,11)(8,16)(9,14)(10,18)(12,19)(13,17)(15,20)(21,23)(22,28)(24,34)(25,33)(26,32)(27,31)(29,36)(30,35);(6,12)(10,15)(11,19)(18,20)(22,29)(23,32)(27,30)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh,g^c=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100911);