Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,32,30,26)(24,34,35,33)(25,29,28,36);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,22,27,30,32,31)(24,34,29,25,35,33,36,28);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,26,29,27,35,32,36)(22,28,30,34,31,25,23,33);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,28)(22,29,31,36)(23,24,30,35)(26,33,32,34);(8,15)(11,18)(14,19)(17,20)(24,35)(25,28)(29,36)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473558);