Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C2×#3277 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2013 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,20,35,19,24,18,25)(15,22,30,21,16,28,27,38)(17,31,36,41,34,23,33,43)(26,40,42,44,29,39,32,37);(6,9)(7,11)(13,15)(14,21)(16,19)(17,32)(18,30)(20,27)(22,35)(23,39)(24,38)(25,28)(26,33)(29,36)(31,40)(34,42)(37,41)(43,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,15)(14,22,24,28)(17,26,34,29)(18,30,20,27)(21,25,38,35)(23,40,31,39)(32,36,42,33)(37,43,44,41);(6,9)(10,12)(13,17,18,33,19,34,20,36)(14,23,25,41,24,31,35,43)(15,26,27,32,16,29,30,42)(21,37,22,39,38,44,28,40);(5,7,8,11)(6,10,9,12)(14,24)(15,16)(23,31)(25,35)(26,29)(27,30)(32,42)(41,43)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063356);