Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3547 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,19,24)(15,25,28,38)(16,21,30,36)(17,23,18,22)(20,32,34,27)(26,39,41,43)(29,37,31,35)(33,40,44,42);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,24,34)(16,26,30,41)(17,27,18,32)(21,33,36,44)(22,25,23,38)(29,40,31,42)(35,43,37,39);(2,4)(6,9)(10,12)(13,16,19,30)(14,21,24,36)(15,26,28,41)(17,31,18,29)(20,33,34,44)(22,37,23,35)(25,39,38,43)(27,42,32,40);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,29,30,31)(20,25,34,38)(21,35,36,37)(26,40,41,42)(33,43,44,39);(7,11)(10,12)(16,30)(21,36)(26,41)(29,31)(33,44)(35,37)(39,43)(40,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^c=ei,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6440733);