Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| 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×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,8,11,10)(13,14,17,22,18,23,32,33)(15,19,26,36,25,35,30,20)(16,27,28,38,29,39,31,21)(24,37,41,43,40,44,42,34);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15)(14,19)(16,24)(17,26)(18,25)(20,33)(21,34)(22,36)(23,35)(27,37)(28,41)(29,40)(30,32)(31,42)(38,43)(39,44);(2,4)(5,8,9,10)(6,11,12,7)(13,15)(14,20)(16,24)(17,30)(18,25)(19,33)(21,37)(22,35)(23,36)(26,32)(27,34)(28,42)(29,40)(31,41)(38,44)(39,43);(6,12)(7,11)(13,16,18,29)(14,21,23,38)(15,24,25,40)(17,31,32,28)(19,34,35,43)(20,37,36,44)(22,39,33,27)(26,42,30,41);(15,25)(19,35)(20,36)(24,40)(26,30)(34,43)(37,44)(41,42)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411011);