Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #35571 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,26,28)(23,30,24,34)(25,32,27,31)(29,35,36,33);(2,3)(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(22,28)(23,24)(31,32)(33,35);(1,2,4,3)(21,23,26,24)(22,29,28,36)(25,33,27,35)(30,32,34,31);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,26,23)(22,30,28,34)(25,33,27,35)(29,32,36,31);(1,3,4,2)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,26,27)(22,31,28,32)(23,33,24,35)(29,30,36,34)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719236);