Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535641 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,24,31,25,30,33,34)(22,26,27,29,28,35,32,36);(1,2,3,4)(7,13)(10,16)(14,19)(17,20);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(23,30)(26,35)(29,36)(31,34);(8,15)(11,18)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476813);