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 | #36906 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,18,22,25)(19,23,27,20)(21,24,30,32)(26,31,29,28)(33,34)(35,44)(36,37)(38,46)(39,40)(41,45)(42,48)(43,47);(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32)(34,40)(35,45)(37,42)(43,46);(1,3,5,11)(2,6,9,7)(4,10,13,16)(8,14,12,15)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,36)(34,42)(35,46)(37,40)(38,41)(39,48)(43,45)(44,47);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,37,39,42)(34,36,40,48)(35,47,45,38)(41,46,44,43);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38,39,47)(34,43,40,46)(35,42,45,37)(36,41,48,44)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=hi,e^2=g,f^2=g,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=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8757764);