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 | #59674 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(17,18)(19,26)(20,23)(21,25)(22,28)(24,29)(27,31)(30,32)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19)(18,22)(20,27)(21,28)(23,30)(24,31)(25,26)(29,32)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,20,21,29)(18,23,25,24)(19,27,28,32)(22,30,26,31)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,24)(19,27)(21,29)(22,31)(23,25)(26,30)(28,32)(33,35,38,43)(34,36,40,41)(37,42,46,48)(39,44,45,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,18)(19,22)(20,23)(21,25)(24,29)(26,28)(27,30)(31,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,19)(18,22)(20,27)(21,28)(23,30)(24,31)(25,26)(29,32)(33,37)(34,39)(35,42)(36,44)(38,46)(40,45)(41,47)(43,48)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8072549);