Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4186 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)(13,14,19,23)(15,24,27,34)(16,21,30,38)(17,22,18,31)(20,32,35,26)(25,39,42,43)(28,36,29,37)(33,40,44,41);(5,7)(6,9)(8,11)(10,12)(13,15,19,27)(14,20,23,35)(16,25,30,42)(17,26,18,32)(21,33,38,44)(22,34,31,24)(28,40,29,41)(36,39,37,43);(2,4)(7,11)(9,12)(13,16)(14,21)(15,25)(17,29)(18,28)(19,30)(20,33)(22,37)(23,38)(24,39)(26,41)(27,42)(31,36)(32,40)(34,43)(35,44);(5,7)(6,9)(8,11)(10,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,28,30,29)(20,34,35,24)(21,36,38,37)(25,40,42,41)(33,39,44,43);(7,11)(9,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,30,28)(20,34,35,24)(21,37,38,36)(25,41,42,40)(33,43,44,39)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6498743);