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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1698 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)(22,27)(24,25)(28,34)(29,33)(31,32)(35,36);(2,3)(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(23,30)(28,35)(31,32)(34,36);(1,2,4,3)(21,22,26,29)(23,28,30,35)(24,27,25,33)(31,36,32,34);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,23,26,30)(22,28,29,35)(24,32,25,31)(27,34,33,36);(1,3,4,2)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,26,25)(22,27,29,33)(23,31,30,32)(28,36,35,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=hi,d^2=fi,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, 6719234);