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 | |
4063 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)(34,40)(35,37)(36,44)(43,48);(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)(33,34,39,40)(35,41,37,47)(36,45,44,38)(42,43,46,48);(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)(33,35,39,37)(34,41,40,47)(36,42,44,46)(38,48,45,43);(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)(34,38)(35,42)(37,46)(39,44)(40,45)(41,48)(43,47);(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)(33,37,39,35)(34,41,40,47)(36,46,44,42)(38,48,45,43);(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)(34,36)(35,43)(37,48)(39,45)(40,44)(41,46)(42,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=i,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, 8179096);