Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533075 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23591 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18)(19,29)(20,34)(21,27)(22,28)(23,26)(24,32)(25,35)(30,38)(31,39)(33,42)(36,41)(37,40)(43,44)(45,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,39)(27,29)(28,42)(34,47)(35,44)(36,46)(37,45)(40,48)(41,43);(2,6)(7,13)(8,14)(15,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,40)(27,41)(28,34)(29,43)(31,44)(32,45)(33,46)(39,48)(42,47);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(18,26)(24,39)(25,40)(27,28)(29,42)(34,41)(38,48)(43,47);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21,23,22)(18,27,26,28)(19,31,33,32)(20,35,36,37)(24,29,39,42)(25,41,40,34)(30,44,46,45)(38,43,48,47)Polycyclic presentation:
2^9|d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6442081);