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 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(10,14)(15,16)(17,18,22,26)(19,27,28,23)(20,24,31,38)(21,34,35,25)(29,39,32,36)(30,37,42,41)(33,46,44,40)(43,47,45,48);(9,10,13,14)(11,15,12,16)(19,28)(20,31)(23,27)(24,38)(30,42)(33,44)(37,41)(40,46);(2,5)(6,7)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,32)(21,30)(22,28)(24,39)(25,37)(26,27)(29,31)(33,45)(34,41)(35,42)(36,38)(40,48)(43,44)(46,47);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,20)(18,24)(19,29)(21,33)(22,31)(23,36)(25,40)(26,38)(27,39)(28,32)(30,43)(34,46)(35,44)(37,47)(41,48)(42,45);(1,3)(2,6)(4,8)(5,7)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,39)(30,41)(31,38)(32,36)(33,40)(34,35)(37,42)(43,48)(44,46)(45,47);(17,21,22,35)(18,25,26,34)(19,30,28,42)(20,33,31,44)(23,37,27,41)(24,40,38,46)(29,43,32,45)(36,47,39,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622628);