Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 | |
2039 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)(15,22,29,35)(16,21,28,36)(17,23,33,40)(18,24,19,25)(27,37,42,43)(30,34,41,44)(31,38,32,39);(5,7)(6,9)(8,11)(10,12)(13,15,19,28,20,29,18,16)(14,21,24,35,26,36,25,22)(17,27,31,41,33,42,32,30)(23,34,39,43,40,44,38,37);(2,4)(7,11)(9,12)(13,16,18,29,20,28,19,15)(14,22,25,36,26,35,24,21)(17,30,32,42,33,41,31,27)(23,37,38,44,40,43,39,34);(7,11)(9,12)(13,17)(14,23)(15,27)(16,30)(18,32)(19,31)(20,33)(21,34)(22,37)(24,39)(25,38)(26,40)(28,41)(29,42)(35,43)(36,44);(6,10)(9,12)(13,18,20,19)(14,24,26,25)(15,16,29,28)(17,31,33,32)(21,35,36,22)(23,38,40,39)(27,41,42,30)(34,37,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=i,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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378138);