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 | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(17,18)(19,25)(20,21)(22,30)(23,24)(26,31)(27,28)(29,32)(33,34,38,40)(35,41,44,39)(36,45,46,37)(42,43,48,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(18,24)(20,25)(26,30)(28,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,41)(45,47)(46,48);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,19,23,21)(18,20,24,25)(22,27,31,29)(26,28,30,32)(33,34,38,40)(35,39,44,41)(36,45,46,37)(42,47,48,43);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32)(33,36,38,46)(34,37,40,45)(35,42,44,48)(39,43,41,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,37)(34,36)(35,43)(38,45)(39,42)(40,46)(41,48)(44,47);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,41)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,d^2=gh,e^2=h,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, 8064478);