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 | |
4229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18,23,25)(19,24,21,20)(22,26,31,32)(27,30,29,28)(33,34,38,39)(35,42,37,40)(36,44,46,41)(43,47,45,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,30,32,28)(34,39)(40,42)(41,44)(47,48);(4,11)(7,9)(12,16)(14,15)(18,25)(19,21)(26,32)(27,29)(33,35,38,37)(34,40,39,42)(36,45,46,43)(41,48,44,47);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20,23,24)(18,19,25,21)(22,30,31,28)(26,29,32,27)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,30,32,28)(33,36,38,46)(34,41,39,44)(35,43,37,45)(40,47,42,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,37,38,35)(34,40,39,42)(36,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441871);