Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,24)(16,30)(17,25)(18,36)(19,23)(20,26)(21,28)(22,34)(27,43)(29,38)(31,41)(32,40)(33,37)(35,42)(39,44);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,28,20,29,17,18)(14,21,25,24,26,36,23,38)(16,31,32,33,34,27,35,44)(22,39,40,43,30,37,42,41);(6,10)(7,11)(13,16)(14,22)(15,27)(17,35)(18,33)(19,32)(20,34)(21,37)(23,42)(24,41)(25,40)(26,30)(28,44)(29,31)(36,39)(38,43);(2,4)(6,10)(9,12)(13,17,20,19)(14,23,26,25)(15,18,29,28)(16,32,34,35)(21,38,36,24)(22,40,30,42)(27,44,31,33)(37,41,39,43);(13,18,17,29,20,28,19,15)(14,24,23,21,26,38,25,36)(16,33,35,31,34,44,32,27)(22,41,42,37,30,43,40,39)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=i,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434588);