Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2043 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)(5,6,8,10)(7,9,11,12)(13,14)(15,24)(16,21)(17,22)(18,33)(19,23)(20,28)(25,39)(26,40)(27,35)(29,43)(30,37)(31,38)(32,44)(34,42)(36,41);(5,7)(6,9)(8,11)(10,12)(13,15,17,27,19,26,18,28)(14,20,33,40,23,35,22,24)(16,29,31,42,30,25,32,41)(21,36,44,39,37,34,38,43);(2,4)(7,11)(9,12)(13,16)(14,21)(15,25)(17,31)(18,32)(19,30)(20,34)(22,38)(23,37)(24,39)(26,29)(27,41)(28,42)(33,44)(35,36)(40,43);(5,7)(6,9)(8,11)(10,12)(15,26)(16,30)(20,35)(21,37)(24,40)(27,28)(31,32)(38,44);(7,11)(9,12)(13,17,19,18)(14,22,23,33)(15,27,26,28)(16,31,30,32)(20,24,35,40)(21,38,37,44)(25,41,29,42)(34,39,36,43)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6499067);