Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22,25,27)(23,31,32,28)(24,29,33,35)(26,30,34,36);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(22,27)(23,24)(26,34)(28,29)(31,35)(32,33);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23,25,32)(22,28,27,31)(24,26,33,34)(29,36,35,30);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(22,27)(23,32)(24,33)(30,36);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,24)(22,29)(23,26)(25,33)(27,35)(28,36)(30,31)(32,34)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6595047);