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 | |
4267 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,12,13,7)(10,15,16,14)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(36,44)(40,47)(42,48)(45,46);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(18,24)(19,20)(26,29)(27,28)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,47)(45,48);(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,45)(44,47)(46,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8042910);