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 | |
4062 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,15)(13,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3,5,11)(2,6,7,9)(4,13,14,10)(8,16,12,15)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(34,35)(39,41)(40,42)(46,47);(2,7)(6,9)(8,12)(15,16)(21,29)(24,31)(26,32)(28,30)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,5,14)(2,8,7,12)(3,10,11,13)(6,15,9,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8177764);