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 | |
4058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(8,13)(9,15)(14,16)(17,18,22,24)(19,23,26,20)(21,29,30,25)(27,28,32,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,47)(45,46);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,19)(18,23)(20,24)(21,27)(22,26)(25,31)(28,29)(30,32)(33,35,39,36)(34,40,43,41)(37,44,42,46)(38,45,48,47);(2,6)(7,12)(8,13)(14,16)(18,24)(19,26)(25,29)(27,32)(33,35,39,36)(34,41,43,40)(37,46,42,44)(38,45,48,47);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36,39,35)(34,41,43,40)(37,46,42,44)(38,47,48,45);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,37,39,42)(34,38,43,48)(35,44,36,46)(40,45,41,47);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,21,22,30)(18,25,24,29)(19,27,26,32)(20,28,23,31)(33,38)(34,42)(35,45)(36,47)(37,43)(39,48)(40,46)(41,44)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025806);