Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | |
2095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,36,19,25,18,24)(15,26,32,21,28,42,29,40)(16,30,37,39,33,22,34,44)(17,23,31,43,35,38,27,41);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,25,40)(16,31,33,27)(17,34,35,37)(18,32,20,29)(22,23,30,38)(24,42,36,26)(39,43,44,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,33)(14,22,25,30)(15,27,28,31)(17,29,35,32)(18,34,20,37)(21,38,40,23)(24,39,36,44)(26,43,42,41);(7,12)(9,11)(13,17,20,31,19,35,18,27)(14,23,36,43,25,38,24,41)(15,16,32,37,28,33,29,34)(21,39,42,22,40,44,26,30);(6,10)(9,11)(15,28)(16,33)(21,40)(22,30)(26,42)(29,32)(34,37)(39,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,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, 6914081);