Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 | |
2242 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)(7,8)(9,11)(10,12)(13,14)(15,23)(16,27)(17,30)(18,21)(19,26)(20,28)(22,33)(24,34)(25,37)(29,38)(31,35)(32,36)(39,41)(40,44)(42,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12);(2,4)(5,8,9,10)(6,7,11,12)(13,15)(14,19)(16,24)(17,31)(18,25)(20,34)(21,35)(22,37)(23,30)(26,33)(27,39)(28,42)(29,40)(32,41)(36,44)(38,43);(6,11)(7,12)(13,16,18,29)(14,20,21,36)(15,24,25,40)(17,32,33,28)(19,34,35,44)(22,38,30,27)(23,39,37,43)(26,42,31,41);(14,21)(15,25)(20,36)(22,30)(24,40)(26,31)(27,38)(41,42)Polycyclic presentation:
2^9|c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408507);