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 | #37483 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4168 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)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(2,6)(7,9)(8,12)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(3,10)(7,9)(11,13)(15,16)(18,22)(23,25)(24,28)(30,31)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,14)(12,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=hi,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=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297738);