Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | #24362 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2021 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,20)(18,22)(23,39)(24,33)(25,34)(26,32)(27,37)(28,35)(29,38)(30,36)(31,40)(41,44)(42,43);(2,4)(13,15)(14,19)(16,27)(17,25)(18,23)(20,34)(21,37)(22,39)(24,29)(26,30)(28,42)(31,41)(32,36)(33,38)(35,43)(40,44);(6,9)(10,12)(14,20)(15,23)(18,30)(19,32)(21,35)(24,27)(25,26)(29,31)(33,43)(34,39)(37,44)(41,42);(13,16,17,28)(14,21,20,35)(15,24,25,41)(18,31,30,29)(19,33,34,44)(22,40,36,38)(23,27,26,42)(32,43,39,37);(5,7,8,11)(6,10,9,12)(14,20)(19,34)(21,35)(22,36)(32,39)(33,44)(37,43)(38,40)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361370);