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 | |
4138 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)(3,8)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,34,40,44)(35,43,39,36)(37,46,48,41)(38,45,47,42);(2,5)(8,10)(9,13)(15,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,35,40,39)(34,36,44,43)(37,45,48,42)(38,41,47,46);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,36)(34,39)(35,44)(37,47)(38,48)(40,43)(41,42)(45,46);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)(33,37,40,48)(34,41,44,46)(35,45,39,42)(36,47,43,38);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,38,40,47)(34,42,44,45)(35,46,39,41)(36,48,43,37);(1,6,7,14)(2,9,5,13)(3,11,4,12)(8,15,10,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,39,40,35)(34,43,44,36)(37,45,48,42)(38,41,47,46)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=g,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=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315565);