Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,4)(6,7)(9,10,13,15)(11,12,16,14)(17,18,23,28)(19,24,29,33)(20,32,35,25)(21,26,36,30)(22,27,39,44)(31,40,38,43)(34,46,47,41)(37,42,48,45);(1,2,5,8)(3,6,4,7)(9,11,13,16)(10,14,15,12)(19,29)(21,36)(24,33)(26,30)(31,38)(37,48)(40,43)(42,45);(1,3,5,4)(2,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,30)(21,32)(22,38)(23,29)(25,36)(26,35)(27,43)(28,33)(31,39)(34,42)(37,41)(40,44)(45,47)(46,48);(9,12)(10,11)(13,14)(15,16)(17,20,23,35)(18,25,28,32)(19,30,29,26)(21,24,36,33)(22,34,39,47)(27,41,44,46)(31,45,38,42)(37,40,48,43);(1,2)(3,6)(4,7)(5,8)(17,21,23,36)(18,26,28,30)(19,25,29,32)(20,33,35,24)(22,37,39,48)(27,42,44,45)(31,41,38,46)(34,43,47,40);(1,4,5,3)(2,7,8,6)(17,22)(18,27)(19,31)(20,34)(21,37)(23,39)(24,40)(25,41)(26,42)(28,44)(29,38)(30,45)(32,46)(33,43)(35,47)(36,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947301);