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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,8)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,34)(35,44)(36,37)(38,46)(39,40)(41,45)(42,48)(43,47);(1,3,6,8)(2,4,7,10)(5,11,14,15)(9,12,13,16)(17,18,22,24)(19,20,25,27)(21,23,29,30)(26,28,31,32)(34,40)(35,45)(37,42)(43,46);(2,7)(4,10)(5,14)(11,15)(19,25)(20,27)(21,29)(23,30)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,2)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,19,22,25)(18,20,24,27)(21,26,29,31)(23,28,30,32)(33,36)(34,42)(35,46)(37,40)(38,41)(39,48)(43,45)(44,47);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,20)(18,19)(21,28)(22,27)(23,26)(24,25)(29,32)(30,31)(33,37,39,42)(34,36,40,48)(35,47,45,38)(41,46,44,43);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,21,22,29)(18,23,24,30)(19,26,25,31)(20,28,27,32)(33,38,39,47)(34,43,40,46)(35,42,45,37)(36,41,48,44)Polycyclic presentation:
2^9|b^2=gi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961191);