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 | |
4147 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)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,18,23,24)(19,25,20,29)(21,22,26,32)(27,31,30,28)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(18,24)(21,26)(25,29)(27,30)(36,44)(40,46)(43,47)(45,48);(2,6)(7,9)(8,12)(14,15)(17,19,23,20)(18,25,24,29)(21,27,26,30)(22,31,32,28)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,23,19)(18,25,24,29)(21,30,26,27)(22,31,32,28)(33,36,39,44)(34,40,38,46)(35,43,41,47)(37,45,42,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,23,26)(18,22,24,32)(19,27,20,30)(25,31,29,28)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,22)(18,26)(19,28)(20,31)(21,24)(23,32)(25,27)(29,30)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,d^2=gh,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, 8161070);