Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2872 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)(7,9)(8,10)(11,12)(13,14,18,23,19,24,30,37)(15,25,28,36,29,40,33,20)(16,21,17,22,31,38,32,39)(26,41,43,44,42,35,27,34);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,26)(17,27)(18,33)(19,29)(21,34)(22,35)(23,40)(24,36)(25,37)(28,30)(31,42)(32,43)(38,44)(39,41);(2,4)(6,10)(9,12)(13,16,19,31)(14,21,24,38)(15,26,29,42)(17,30,32,18)(20,34,36,44)(22,37,39,23)(25,41,40,35)(27,28,43,33);(2,4)(13,17)(14,22)(15,27)(16,30)(18,31)(19,32)(20,35)(21,37)(23,38)(24,39)(25,34)(26,28)(29,43)(33,42)(36,41)(40,44);(16,31)(17,32)(21,38)(22,39)(26,42)(27,43)(34,44)(35,41)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347781);