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 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4140 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,24,28)(19,29,31,25)(20,22,21,26)(23,27,32,30)(33,34)(35,36)(37,45)(38,40)(39,47)(41,43)(42,48)(44,46);(2,6)(7,9)(8,13)(14,15)(17,19,24,31)(18,25,28,29)(20,30,21,27)(22,32,26,23)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(4,12)(8,13)(10,16)(14,15)(17,20,24,21)(18,26,28,22)(19,30,31,27)(23,29,32,25)(33,34,38,40)(35,41,43,36)(37,39,47,45)(42,48,46,44);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,21,24,20)(18,22,28,26)(19,30,31,27)(23,29,32,25)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,22)(18,20)(19,23)(21,28)(24,26)(25,27)(29,30)(31,32)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,23)(18,27)(19,22)(20,25)(21,29)(24,32)(26,31)(28,30)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=g,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, 8109177);