Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,30)(17,23)(18,24)(19,34)(20,25)(22,31)(26,41)(27,36)(28,37)(29,39)(32,40)(33,38)(35,43)(42,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,20,29)(14,21,25,39)(16,26,17,27)(18,33,19,28)(22,35,40,44)(23,36,30,41)(24,38,34,37)(31,43,32,42);(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,26)(17,20)(18,31)(19,32)(21,35)(23,24)(25,40)(27,29)(28,42)(30,34)(33,43)(36,38)(37,41)(39,44);(1,3,4,2)(7,12)(9,11)(13,17,20,16)(14,23,25,30)(15,27,29,26)(18,31,19,32)(21,36,39,41)(22,34,40,24)(28,42,33,43)(35,37,44,38);(6,10)(9,11)(13,18,20,19)(14,24,25,34)(15,28,29,33)(16,31,17,32)(21,37,39,38)(22,23,40,30)(26,42,27,43)(35,41,44,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903782);