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 | |
4116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,22,23)(19,26,24,20)(21,29,30,25)(27,28,31,32)(33,34)(35,38)(36,44)(37,41)(39,43)(40,46)(42,48)(45,47);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(18,23)(20,26)(21,30)(27,31)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(2,6)(7,10)(8,12)(14,16)(17,19,22,24)(18,20,23,26)(21,27,30,31)(25,28,29,32)(33,34,37,41)(35,39,43,38)(36,40,46,44)(42,48,45,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,29)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,30)(18,25,23,29)(19,27,24,31)(20,28,26,32)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8762624);