Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,11)(7,10,8,12)(13,14,20,26,19,25,18,24)(15,27,29,21,30,38,17,23)(16,31,36,41,35,22,34,42)(28,39,32,40,44,43,33,37);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,32)(17,20)(18,29)(19,30)(22,39)(23,25)(24,38)(26,27)(28,36)(31,43)(33,35)(34,44)(37,42)(40,41);(2,4)(5,8,9,7)(6,10,11,12)(13,16,20,36,19,35,18,34)(14,22,26,42,25,31,24,41)(15,28,29,32,30,44,17,33)(21,37,38,39,23,40,27,43);(2,4)(13,17,18,30,19,29,20,15)(14,23,24,38,25,21,26,27)(16,33,34,44,35,32,36,28)(22,40,41,39,31,37,42,43);(13,18,19,20)(14,24,25,26)(15,29,30,17)(16,34,35,36)(21,38,23,27)(22,41,31,42)(28,32,44,33)(37,39,40,43)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355345);