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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3143 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)(13,14)(15,25)(16,30)(17,23)(18,24)(19,22)(20,29)(21,31)(26,40)(27,35)(28,36)(32,37)(33,38)(34,41)(39,43)(42,44);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,26)(17,28)(18,29)(19,27)(21,34)(22,36)(23,25)(24,35)(30,44)(31,42)(32,43)(33,41)(37,40)(38,39);(6,9)(7,11)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,37)(23,38)(24,30)(25,39)(27,41)(28,42)(29,43)(35,44)(36,40);(5,7)(6,10)(8,11)(9,12)(14,22)(15,27)(21,37)(23,24)(26,41)(28,29)(30,38)(42,43);(6,9)(7,11)(13,17,19,18)(14,23,22,24)(15,28,27,29)(16,31,33,32)(20,25,36,35)(21,38,37,30)(26,42,41,43)(34,39,40,44)Polycyclic presentation:
2^9|b^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503133);