Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,10,11,12)(13,14,17,22)(15,24,26,19)(16,21,28,35)(18,20,30,23)(25,37,38,32)(27,33,40,31)(29,34,41,36)(39,43,44,42);(1,2)(3,4)(6,9)(10,12)(13,15,17,26)(14,19,22,24)(16,25,28,38)(18,27,30,40)(20,31,23,33)(21,32,35,37)(29,39,41,44)(34,42,36,43);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(18,30)(19,31)(21,34)(22,23)(24,33)(27,40)(29,41)(32,42)(35,36)(37,43)(39,44);(2,4)(13,16)(14,21)(15,25)(17,28)(18,29)(19,32)(20,34)(22,35)(23,36)(24,37)(26,38)(27,39)(30,41)(31,42)(33,43)(40,44);(14,22)(15,26)(16,28)(20,23)(27,40)(29,41)(32,37)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345785);