Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,15)(7,11,17,20)(8,14,9,13)(12,18,19,16)(21,22,25,29,26,23,27,31)(24,28,33,35,32,30,34,36);(1,2,4,3)(5,7)(6,11)(8,18)(9,16)(10,17)(12,13)(14,19)(15,20)(21,23,25,31,26,22,27,29)(24,30,33,36,32,28,34,35);(1,3,4,2)(5,7)(6,12)(8,16)(9,18)(10,17)(11,13)(14,20)(15,19)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,8,10,9)(6,13,15,14)(7,16,17,18)(11,19,20,12)(22,23)(24,32)(29,31)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,d^b=dhi,d^c=di,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109671);