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 | #40720 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4264 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,31)(22,27)(23,25)(24,29)(26,28)(30,32)(33,34,39,42)(35,43,37,40)(36,38,41,48)(44,47,46,45);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,29)(18,24,25,20)(21,22,28,32)(26,27,31,30)(33,35,39,37)(34,40,42,43)(36,44,41,46)(38,45,48,47);(3,10)(6,7)(11,16)(14,15)(18,25)(20,24)(26,31)(27,30)(33,36,39,41)(34,38,42,48)(35,44,37,46)(40,45,43,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,24)(18,19,25,29)(21,30,28,27)(22,31,32,26)(33,36)(34,38)(35,44)(37,46)(39,41)(40,45)(42,48)(43,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,26)(19,22)(20,30)(23,28)(24,27)(25,31)(29,32)(33,37,39,35)(34,40,42,43)(36,46,41,44)(38,45,48,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,31)(25,30)(33,38)(34,41)(35,45)(36,42)(37,47)(39,48)(40,46)(43,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635699);