Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14,19,24)(15,20,28,37)(16,21,17,22)(18,23,33,42)(25,34,30,39)(26,35,43,44)(27,36,32,41)(29,38,31,40);(2,3)(6,10)(11,12)(13,15,19,28)(14,20,24,37)(16,25,17,30)(18,32,33,27)(21,34,22,39)(23,41,42,36)(26,29,43,31)(35,38,44,40);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,25)(17,19)(18,29)(20,34)(22,24)(23,38)(26,27)(28,30)(31,33)(32,43)(35,36)(37,39)(40,42)(41,44);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,19,16)(14,22,24,21)(15,26,28,43)(18,29,33,31)(20,35,37,44)(23,38,42,40)(25,32,30,27)(34,41,39,36);(6,10)(11,12)(16,17)(21,22)(25,30)(26,43)(29,31)(34,39)(35,44)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468829);