Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,21)(17,22)(18,33)(19,23)(20,27)(25,35)(26,40)(28,41)(29,36)(30,44)(31,38)(32,39)(34,43)(37,42);(1,2)(3,4)(13,15,18,27,19,28,16,25)(14,20,21,24,23,35,33,41)(17,30,31,42,32,26,29,43)(22,37,36,44,39,34,38,40);(2,4)(6,9)(10,12)(13,16,19,18)(14,21,23,33)(15,25,28,27)(17,31,32,29)(20,24,35,41)(22,38,39,36)(26,43,30,42)(34,44,37,40);(2,4)(13,17)(14,22)(15,26)(16,29)(18,31)(19,32)(20,34)(21,36)(23,39)(24,40)(25,42)(27,43)(28,30)(33,38)(35,37)(41,44);(5,7,8,11)(6,10,9,12)(17,32)(22,39)(26,30)(29,31)(34,37)(36,38)(40,44)(42,43)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361818);