Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,27,36)(16,21,30,40)(17,31,18,22)(24,33,26,35)(25,34,41,44)(28,37,42,43)(29,39,32,38);(2,3)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,28)(17,26)(18,24)(19,27)(21,37)(22,33)(23,36)(25,29)(30,42)(31,35)(32,41)(34,39)(38,44)(40,43);(1,2,4,3)(6,10)(9,12)(15,24)(17,18)(20,33)(22,31)(25,28)(26,27)(29,32)(34,37)(35,36)(38,39)(41,42)(43,44);(1,3,4,2)(13,16)(14,21)(15,25)(17,29)(18,32)(19,30)(20,34)(22,38)(23,40)(24,28)(26,42)(27,41)(31,39)(33,37)(35,43)(36,44);(13,17,19,18)(14,22,23,31)(15,24,27,26)(16,29,30,32)(20,35,36,33)(21,38,40,39)(25,28,41,42)(34,43,44,37)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348881);