Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3168 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)(13,14)(15,24)(16,21)(17,22)(18,33)(19,23)(20,27)(25,41)(26,35)(28,40)(29,44)(30,37)(31,38)(32,39)(34,42)(36,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,23,35)(16,29,30,25)(17,27,18,26)(21,36,37,34)(22,40,33,24)(31,42,32,43)(38,44,39,41);(2,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,32)(18,31)(19,30)(20,34)(22,39)(23,37)(24,41)(26,43)(27,42)(28,29)(33,38)(35,36)(40,44);(5,7,8,11)(6,10,9,12)(16,30)(21,37)(25,29)(31,32)(34,36)(38,39)(41,44)(42,43);(6,9)(10,12)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,31,30,32)(20,24,35,40)(21,38,37,39)(25,42,29,43)(34,44,36,41)Polycyclic presentation:
2^9|b^2=i,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419856);