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 | #3493 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,24,29)(19,30,25,35)(20,33,36,32)(21,37,40,26)(22,27,23,28)(31,43,42,46)(34,47,48,38)(39,45,41,44);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,24,25)(18,20,29,36)(21,38,40,47)(22,32,23,33)(26,43,37,46)(27,35,28,30)(31,41,42,39)(34,44,48,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,25)(19,29)(21,34)(22,35)(23,30)(24,36)(26,42)(27,33)(28,32)(31,37)(38,44)(39,43)(40,48)(41,46)(45,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21)(18,26)(19,31)(20,34)(22,39)(23,41)(24,40)(25,42)(27,44)(28,45)(29,37)(30,46)(32,47)(33,38)(35,43)(36,48);(2,8)(6,9)(7,16)(14,15)(17,22,24,23)(18,27,29,28)(19,32,25,33)(20,35,36,30)(21,39,40,41)(26,44,37,45)(31,47,42,38)(34,43,48,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=i,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, 7031315);