Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,19,23)(15,20,27,36)(16,28,31,40)(17,24,18,22)(21,33,38,30)(25,41,29,44)(26,39,32,35)(34,42,37,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,29)(17,32)(18,26)(19,27)(21,37)(22,39)(23,36)(24,35)(25,31)(28,44)(30,42)(33,43)(34,38)(40,41);(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,33)(19,31)(20,34)(22,28)(23,38)(24,40)(26,42)(27,29)(32,43)(35,44)(36,37)(39,41);(6,9)(10,12)(13,17,19,18)(14,22,23,24)(15,26,27,32)(16,30,31,33)(20,35,36,39)(21,28,38,40)(25,42,29,43)(34,44,37,41);(5,7,8,11)(6,10,9,12)(14,23)(15,27)(21,38)(22,24)(25,29)(26,32)(28,40)(42,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398847);