Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2050 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)(7,9)(8,10)(11,12)(13,14,20,26)(15,22,28,36)(16,21,29,35)(17,23,31,38)(18,25,19,24)(27,37,42,43)(30,34,41,44)(32,40,33,39);(5,7)(6,9)(8,11)(10,12)(13,15,19,29,20,28,18,16)(14,21,25,36,26,35,24,22)(17,27,33,41,31,42,32,30)(23,34,40,43,38,44,39,37);(2,4)(7,11)(9,12)(13,16,18,28,20,29,19,15)(14,22,24,35,26,36,25,21)(17,30,32,42,31,41,33,27)(23,37,39,44,38,43,40,34);(7,11)(9,12)(13,17)(14,23)(15,27)(16,30)(18,32)(19,33)(20,31)(21,34)(22,37)(24,39)(25,40)(26,38)(28,42)(29,41)(35,44)(36,43);(6,10)(9,12)(15,28)(16,29)(17,31)(21,35)(22,36)(23,38)(32,33)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378106);