Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3149 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)(7,11)(10,12)(13,14)(15,25)(16,22)(17,24)(18,21)(19,23)(20,28)(26,36)(27,41)(29,35)(30,44)(31,40)(32,39)(33,37)(34,42)(38,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,20,23,35)(16,30,31,27)(17,29,18,28)(21,36,24,25)(22,38,40,34)(32,42,33,43)(37,41,39,44);(6,9)(10,12)(14,21)(15,26)(17,18)(20,25)(22,37)(23,24)(27,30)(32,33)(34,41)(35,36)(38,44)(39,40);(6,9)(10,12)(13,16,19,31)(14,22,23,40)(15,27,26,30)(17,32,18,33)(20,34,35,38)(21,37,24,39)(25,41,36,44)(28,42,29,43);(5,7,8,11)(6,10,9,12)(14,23)(16,31)(20,35)(21,24)(25,36)(27,30)(32,33)(42,43)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=hi,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,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398321);