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 |
| prime radical | #47067 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,14,9)(4,15,5,6)(11,13,12,16)(33,34)(35,42)(36,39)(37,44)(38,40)(41,46)(43,47)(45,48);(1,3,7,14)(2,8,10,9)(4,12,5,11)(6,13,15,16)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(33,35,39,38)(34,40,36,42)(37,45,46,43)(41,48,44,47);(1,4,7,5)(2,6,10,15)(3,11,14,12)(8,16,9,13)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(1,5,7,4)(2,6,10,15)(3,12,14,11)(8,16,9,13)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,3)(2,9)(4,11)(5,12)(6,13)(7,14)(8,10)(15,16)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,6)(2,4)(3,13)(5,10)(7,15)(8,12)(9,11)(14,16)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=gi,e^2=ghi,f^2=hi,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, 8179088);