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 | |
4167 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)(3,4)(5,13)(6,7)(8,11)(9,15)(10,16)(12,14)(17,18,23,26)(19,20,24,21)(22,25,30,32)(27,28,31,29)(33,34)(35,44)(36,42)(37,48)(38,46)(39,43)(40,47)(41,45);(2,7)(4,8)(9,13)(12,16)(17,19,23,24)(18,21,26,20)(22,27,30,31)(25,29,32,28)(33,35)(34,38)(36,37)(39,46)(40,48)(41,42)(43,44)(45,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,16)(12,13)(14,15)(17,20,23,21)(18,24,26,19)(22,28,30,29)(25,31,32,27)(33,36)(34,40)(35,45)(37,46)(38,41)(39,47)(42,43)(44,48);(1,4,6,8)(2,3,7,11)(5,12,15,16)(9,10,13,14)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31)(33,37,39,45)(34,41,43,48)(35,36,46,47)(38,42,44,40);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,21,23,20)(18,19,26,24)(22,28,30,29)(25,31,32,27)(33,36)(34,42)(35,37)(38,41)(39,47)(40,43)(44,48)(45,46);(1,5)(2,9)(3,10)(4,12)(6,15)(7,13)(8,16)(11,14)(17,22,23,30)(18,25,26,32)(19,27,24,31)(20,28,21,29)(33,38)(34,35)(36,41)(37,42)(39,44)(40,45)(43,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10362908);