Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14)(15,24)(16,21)(17,32)(18,23)(19,22)(20,27)(25,41)(26,35)(28,40)(29,44)(30,39)(31,38)(33,37)(34,42)(36,43);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,29)(17,28)(18,27)(19,26)(21,36)(22,40)(23,24)(25,30)(31,43)(32,35)(33,42)(34,39)(37,41)(38,44);(2,4)(5,7,8,11)(6,9,10,12)(13,16,18,31)(14,21,23,38)(15,25,27,42)(17,30,19,33)(20,34,24,41)(22,37,32,39)(26,29,28,43)(35,44,40,36);(2,4)(6,10)(9,12)(13,17,18,19)(14,22,23,32)(15,26,27,28)(16,30,31,33)(20,35,24,40)(21,37,38,39)(25,29,42,43)(34,44,41,36);(14,23)(15,27)(21,38)(22,32)(25,42)(26,28)(29,43)(37,39)Polycyclic presentation:
2^9|a^2=fi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408619);