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 | |
4238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31)(33,34)(35,40)(36,43)(37,41)(38,39)(42,47)(44,48)(45,46);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27)(34,39)(36,40)(37,45)(42,48);(2,6)(7,9)(8,12)(14,15)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32)(33,35,38,43)(34,40,39,36)(37,42,45,48)(41,47,46,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32)(33,36)(34,35)(37,44)(38,40)(39,43)(41,42)(45,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10239361);