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 | |
4172 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(2,6)(7,9)(8,12)(14,15)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,46)(45,48);(2,6)(3,10)(4,13)(14,15)(18,25)(19,20)(26,32)(27,28)(34,38)(35,37)(36,44)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36,39,44)(34,41,38,45)(35,43,37,46)(40,47,42,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,37,39,35)(34,42,38,40)(36,43,44,46)(41,47,45,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,38,39,34)(35,42,37,40)(36,45,44,41)(43,48,46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=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, 9453113);