Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | #60933 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,23,27)(19,30,33,46)(20,35,38,26)(21,29,22,28)(24,39,43,34)(25,40,44,37)(31,41,36,42)(32,45,48,47);(3,9)(4,12)(6,13)(7,11)(17,19)(18,24)(20,36)(21,39)(22,34)(23,33)(25,42)(26,45)(27,43)(28,46)(29,30)(31,38)(32,37)(35,47)(40,48)(41,44);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,23,38)(18,25,27,44)(19,31,33,36)(21,37,22,40)(24,41,43,42)(26,28,35,29)(30,47,46,45)(32,39,48,34);(3,9)(6,13)(10,15)(14,16)(17,20)(18,26)(19,32)(21,40)(22,37)(23,38)(24,42)(25,29)(27,35)(28,44)(30,47)(31,34)(33,48)(36,39)(41,43)(45,46);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(19,33)(25,44)(26,35)(28,29)(31,36)(32,48)(34,39)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7073049);