Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #30512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14)(15,25)(16,30)(17,23)(18,24)(19,35)(20,29)(21,31)(22,32)(26,42)(27,37)(28,39)(33,38)(34,41)(36,43)(40,44);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,27)(17,33)(18,29)(19,28)(21,40)(22,37)(23,39)(24,25)(26,32)(30,42)(31,43)(34,44)(35,38)(36,41);(1,2,4,3)(13,16)(14,21)(15,26)(17,31)(18,32)(19,34)(20,36)(22,23)(24,41)(25,40)(27,29)(28,43)(30,35)(33,44)(37,38)(39,42);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,27)(17,34)(18,32)(19,31)(20,37)(21,23)(24,30)(25,42)(26,29)(28,43)(33,44)(35,41)(36,38)(39,40);(1,3,4,2)(6,9)(10,12)(13,17,18,19)(14,23,24,35)(15,28,29,33)(16,31,32,34)(20,38,25,39)(21,22,41,30)(26,43,27,44)(36,37,40,42)Polycyclic presentation:
2^9|c^2=hi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6715497);