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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,20)(16,21)(17,22)(18,23)(19,24)(25,32)(26,34)(27,35)(28,36)(29,37)(30,38)(31,33)(39,44)(40,42)(41,43);(2,3)(6,10)(11,12)(13,15,19,27)(14,20,24,35)(16,25,30,41)(17,26,18,31)(21,32,38,43)(22,34,23,33)(28,40,29,39)(36,42,37,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,19,30)(14,21,24,38)(15,25,27,41)(17,28,18,29)(20,32,35,43)(22,36,23,37)(26,40,31,39)(33,44,34,42);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17)(14,22)(16,28)(18,19)(21,36)(23,24)(26,31)(29,30)(33,34)(37,38)(39,40)(42,44);(6,10)(11,12)(13,17,19,18)(14,23,24,22)(15,26,27,31)(16,29,30,28)(20,33,35,34)(21,36,38,37)(25,39,41,40)(32,42,43,44)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468688);