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 | |
4167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,8,13,15)(10,12,16,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(1,3,6,11)(2,7,9,4)(5,10,13,16)(8,14,15,12)(18,19)(23,25)(24,26)(30,31)(36,44)(40,46)(43,47)(45,48);(5,13)(8,15)(10,16)(12,14)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,14)(15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,39,44)(34,40,38,46)(35,43,41,47)(37,45,42,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,5,6,13)(2,8,9,15)(3,10,11,16)(4,12,7,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=i,d^2=h,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8167876);