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 | #41723 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,27)(19,26,29,22)(20,24,21,25)(28,30,31,32)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,45)(46,47);(2,7)(3,10)(9,12)(11,16)(17,19,23,29)(18,22,27,26)(20,28,21,31)(24,32,25,30)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,21)(18,24,27,25)(19,28,29,31)(22,32,26,30)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,21,23,20)(18,25,27,24)(19,28,29,31)(22,32,26,30)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,19)(18,26)(20,28)(21,31)(22,27)(23,29)(24,30)(25,32)(33,35,39,38)(34,42,43,40)(36,44,41,46)(37,45,48,47);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22,23,26)(18,19,27,29)(20,30,21,32)(24,31,25,28)(33,38,39,35)(34,40,43,42)(36,46,41,44)(37,47,48,45)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8324204);