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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2156 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)(16,28)(17,33)(18,24)(19,23)(21,27)(22,32)(25,39)(26,38)(29,44)(30,36)(31,41)(34,37)(35,42)(40,43);(2,3)(6,10)(11,12)(13,15,18,27)(14,20,24,21)(16,29,32,43)(17,26,19,34)(22,40,28,44)(23,37,33,38)(25,35,30,31)(36,41,39,42);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,21)(16,30)(17,34)(18,27)(19,26)(20,24)(22,36)(23,37)(25,32)(28,39)(29,35)(31,43)(33,38)(40,42)(41,44);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16)(14,22)(15,25)(17,31)(18,32)(19,35)(20,36)(21,39)(23,41)(24,28)(26,43)(27,30)(29,34)(33,42)(37,40)(38,44);(6,10)(11,12)(13,17,18,19)(14,23,24,33)(15,26,27,34)(16,31,32,35)(20,37,21,38)(22,41,28,42)(25,43,30,29)(36,40,39,44)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fgi,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,d^c=di,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468914);