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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1751 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)(15,23)(16,28)(17,31)(18,21)(19,26)(20,29)(22,32)(24,38)(25,35)(27,34)(30,36)(33,39)(37,40)(41,42)(43,44);(6,9)(7,11)(13,15,18,27)(14,19,21,35)(16,24,29,40)(17,25,32,26)(20,33,28,42)(22,34,31,23)(30,39,44,41)(36,38,43,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,24)(17,30)(18,29)(19,33)(21,28)(22,36)(23,37)(25,39)(26,41)(27,40)(31,43)(32,44)(34,38)(35,42);(6,9)(10,12)(15,25)(17,32)(19,34)(22,31)(23,35)(24,39)(26,27)(30,44)(33,38)(36,43)(37,42)(40,41);(5,7,8,11)(6,10,9,12)(14,21)(16,29)(19,35)(22,31)(23,34)(24,40)(30,44)(39,41)Polycyclic presentation:
2^9|a^2=gi,b^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7055659);