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 | |
4253 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)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32)(33,34)(35,42)(36,44)(37,40)(38,39)(41,46)(43,47)(45,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,34,39,38)(35,40,37,42)(36,41,46,44)(43,47,45,48);(1,2,5,8)(3,6,11,9)(4,12,13,7)(10,15,16,14)(17,20,22,26)(18,23,25,19)(21,28,30,29)(24,31,32,27)(33,35,39,37)(34,40,38,42)(36,45,46,43)(41,48,44,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,20)(18,19)(21,29)(22,26)(23,25)(24,31)(27,32)(28,30)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,37,39,35)(34,40,38,42)(36,43,46,45)(41,48,44,47);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,38,39,34)(35,40,37,42)(36,44,46,41)(43,47,45,48)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831939);