Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60580 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,29)(20,35)(21,28)(22,27)(23,26)(24,33)(25,37)(30,46)(31,42)(32,34)(36,41)(38,40)(39,44)(43,48)(45,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,36)(21,31)(22,33)(23,34)(25,43)(26,42)(27,29)(28,32)(30,37)(35,45)(38,46)(39,47)(40,48)(41,44);(17,20)(18,25)(19,30)(21,37)(22,39)(23,40)(24,41)(26,35)(27,38)(28,44)(29,45)(31,47)(32,46)(33,48)(34,36)(42,43);(2,7)(6,9)(8,15)(14,16)(17,21,22,23)(18,26,27,28)(19,31,33,34)(20,37,39,40)(24,42,29,32)(25,35,38,44)(30,47,48,36)(41,43,45,46);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,18)(19,32)(20,38)(21,26)(22,27)(23,28)(24,31)(25,39)(29,34)(30,43)(33,42)(35,40)(36,41)(37,44)(45,47)(46,48)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580799);