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 | |
4213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,12,15,9)(11,16,13,14)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34,38,39)(35,37,41,47)(36,40,45,48)(42,46,43,44);(2,7)(3,10)(9,12)(11,13)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35)(34,37)(36,43)(38,41)(39,47)(40,44)(42,45)(46,48);(2,7)(5,8)(9,12)(14,16)(20,28)(23,30)(26,32)(29,31)(34,39)(35,41)(40,48)(42,43);(1,3,6,10)(2,8,7,5)(4,13,15,11)(9,14,12,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36,38,45)(34,40,39,48)(35,42,41,43)(37,46,47,44);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,37)(34,35)(36,44)(38,47)(39,41)(40,43)(42,48)(45,46);(1,5,6,8)(2,10,7,3)(4,14,15,16)(9,13,12,11)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,35)(34,37)(36,42)(38,41)(39,47)(40,46)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=gi,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231485);