Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,25,29,40)(16,30,32,21)(17,22,33,39)(18,34,31,23)(20,35,38,28)(26,42,43,37)(27,41,44,36);(2,3)(13,15)(14,20)(16,26)(17,27)(18,35)(19,29)(21,36)(22,37)(23,40)(24,38)(25,34)(28,31)(30,41)(32,43)(33,44)(39,42);(1,2,4,3)(6,9)(10,12)(13,16)(14,21)(15,26)(17,31)(18,33)(19,32)(20,36)(22,34)(23,39)(24,30)(25,37)(27,28)(29,43)(35,44)(38,41)(40,42);(1,3,4,2)(13,17,19,33)(14,22,24,39)(15,27,29,44)(16,31,32,18)(20,37,38,42)(21,34,30,23)(25,41,40,36)(26,28,43,35);(5,7,8,11)(6,10,9,12)(16,32)(17,33)(21,30)(22,39)(26,43)(27,44)(36,41)(37,42)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374417);