Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2997 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)(5,6,8,10)(7,11,12,9)(14,20)(17,18)(21,25)(22,30)(23,24)(28,29)(31,32)(34,39)(35,36)(37,38)(41,42)(43,44);(2,4)(13,14,19,24)(15,25,27,36)(16,30,33,37)(17,23,18,20)(21,28,35,29)(22,31,38,32)(26,34,40,43)(39,41,44,42);(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,26)(17,29)(18,28)(19,27)(20,25)(22,34)(23,36)(24,35)(30,39)(31,42)(32,41)(33,40)(37,44)(38,43);(13,16,19,33)(14,22,24,38)(15,26,27,40)(17,32,18,31)(20,30,23,37)(21,34,35,43)(25,39,36,44)(28,42,29,41);(15,27)(21,35)(25,36)(26,40)(28,29)(34,43)(39,44)(41,42)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352695);