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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 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)(6,10)(8,11)(14,19)(17,31)(20,34)(21,35)(22,32)(23,39)(26,42)(27,38)(29,33)(36,40)(37,44)(41,43);(5,6,9,10)(7,8,12,11)(13,14)(15,23)(16,27)(17,32)(18,22)(19,31)(20,25)(21,29)(24,40)(26,34)(28,44)(30,35)(33,38)(36,43)(37,41)(39,42);(2,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,20)(16,28)(17,26)(18,25)(19,34)(21,37)(22,23)(24,30)(27,40)(29,43)(31,42)(32,39)(33,41)(35,44)(36,38);(6,10)(7,12)(13,16,18,30)(14,21,22,38)(15,24,25,28)(17,33,31,29)(19,35,32,27)(20,36,23,37)(26,43,42,41)(34,40,39,44);(5,8,9,11)(6,7,10,12)(14,22)(15,25)(19,32)(21,38)(24,28)(26,42)(27,35)(41,43)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=i,e^2=hi,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, 7089352);