Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| 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 | #39335 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2192 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)(5,6)(7,10)(8,11)(9,12)(13,14,19,23)(15,24,27,35)(16,21,30,39)(17,22,18,33)(20,31,36,26)(25,41,28,44)(29,38,32,40)(34,42,37,43);(5,7)(6,9)(8,12)(10,11)(13,15,18,26,19,27,17,31)(14,20,22,24,23,36,33,35)(16,28,29,42,30,25,32,43)(21,37,40,44,39,34,38,41);(2,4)(7,12)(9,10)(13,16)(14,21)(15,25)(17,32)(18,29)(19,30)(20,34)(22,40)(23,39)(24,41)(26,43)(27,28)(31,42)(33,38)(35,44)(36,37);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,23,33)(15,26,27,31)(16,29,30,32)(20,35,36,24)(21,38,39,40)(25,42,28,43)(34,41,37,44);(7,12)(9,10)(15,27)(16,30)(20,36)(21,39)(24,35)(26,31)(29,32)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503993);