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 | |
4135 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,26)(22,24)(23,30)(28,32)(29,31)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47);(2,6)(3,9)(8,13)(11,15)(17,19,22,30)(18,23,24,27)(20,28,31,26)(21,29,32,25)(33,35)(34,40)(36,38)(37,47)(39,44)(41,45)(42,46)(43,48);(4,12)(8,13)(11,15)(14,16)(18,24)(21,32)(23,27)(25,29)(35,44)(38,45)(40,48)(42,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,22,31)(18,25,24,29)(19,28,30,26)(21,27,32,23)(33,36)(34,37)(35,45)(38,44)(39,41)(40,42)(43,46)(47,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,19)(18,23)(20,26)(21,29)(22,30)(24,27)(25,32)(28,31)(33,37)(34,41)(35,42)(36,43)(38,40)(39,46)(44,47)(45,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,32)(18,26,24,28)(19,29,30,25)(20,27,31,23)(33,38,39,45)(34,42,43,47)(35,36,44,41)(37,40,46,48)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=gh,f^2=gi,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, 8325348);