Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1798 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)(7,11)(10,12)(13,14)(15,23)(16,21)(17,19)(18,30)(20,26)(22,29)(24,36)(25,35)(27,32)(28,41)(31,39)(33,38)(34,40)(37,43)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(16,27)(20,31)(22,30)(23,35)(24,37)(28,40)(38,44);(6,9)(10,12)(13,15,18,26,17,25,29,39)(14,20,22,23,19,31,30,35)(16,24,28,38,27,37,40,44)(21,33,34,36,32,42,41,43);(6,9)(10,12)(13,16)(14,21)(15,24)(17,27)(18,28)(19,32)(20,33)(22,34)(23,36)(25,37)(26,38)(29,40)(30,41)(31,42)(35,43)(39,44);(5,7,8,11)(6,10,9,12)(15,25)(16,27)(20,31)(21,32)(23,35)(26,39)(28,40)(34,41)Polycyclic presentation:
2^9|a^2=fi,c^2=g,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401341);