Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2036 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)(8,9)(11,12)(13,14)(15,20)(16,29)(17,22)(18,23)(19,24)(21,32)(25,41)(26,36)(27,37)(28,38)(30,44)(31,39)(33,40)(34,42)(35,43);(2,4)(13,15)(14,20)(16,30)(17,25)(18,27)(19,34)(21,36)(22,41)(23,37)(24,42)(26,32)(28,31)(29,44)(33,43)(35,40)(38,39);(6,9)(10,12)(13,16,31,33,18,32,19,17)(14,21,39,22,23,29,24,40)(15,25,34,26,27,43,28,30)(20,35,42,44,37,41,38,36);(13,17,19,32,18,33,31,16)(14,22,24,21,23,40,39,29)(15,26,28,25,27,30,34,43)(20,36,38,41,37,44,42,35);(5,7,8,11)(6,10,9,12)(14,23)(20,37)(21,29)(22,40)(24,39)(35,41)(36,44)(38,42)Polycyclic presentation:
2^9|c^2=hi,d^2=h,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374282);