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 |
| prime radical | #45236 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,22)(15,19,26,35)(16,20,27,36)(17,21,31,40)(23,32,41,43)(24,33,42,44)(25,34,29,38)(28,37,30,39);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,23)(17,29)(18,26)(20,32)(21,38)(22,35)(24,28)(25,31)(27,41)(30,42)(33,37)(34,40)(36,43)(39,44);(2,4)(13,16,18,27)(14,20,22,36)(15,23,26,41)(17,28,31,30)(19,32,35,43)(21,37,40,39)(24,25,42,29)(33,34,44,38);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(14,20)(15,24)(17,30)(18,27)(19,33)(21,39)(22,36)(23,29)(25,41)(26,42)(28,31)(32,38)(34,43)(35,44)(37,40);(6,11)(10,12)(16,27)(20,36)(23,41)(24,42)(28,30)(32,43)(33,44)(37,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=f,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, 7424644);