Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1944 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)(13,14,18,23)(15,25,26,20)(16,28,30,43)(17,32,33,42)(19,35,34,24)(21,37,40,31)(22,38,39,29)(27,41,36,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,29)(17,31)(18,26)(19,36)(21,42)(22,43)(23,25)(24,44)(27,34)(28,39)(30,38)(32,40)(33,37)(35,41);(5,7,8,11)(6,10,9,12)(13,16,18,30)(14,21,23,40)(15,17,26,33)(19,37,34,31)(20,39,25,22)(24,43,35,28)(27,38,36,29)(32,44,42,41);(2,4)(13,17,18,33)(14,22,23,39)(15,16,26,30)(19,38,34,29)(20,40,25,21)(24,42,35,32)(27,37,36,31)(28,44,43,41);(6,9)(10,12)(14,23)(15,26)(16,30)(22,39)(24,35)(27,36)(31,37)(32,42)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422330);