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 | |
4375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,4,5)(6,8,15,16)(11,14,12,13)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(33,34)(35,37)(36,39)(38,44)(40,42)(41,46)(43,47)(45,48);(1,3,7,4)(2,5,9,10)(6,11,15,12)(8,13,16,14)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(33,35)(34,40)(36,37)(38,43)(39,42)(41,47)(44,45)(46,48);(18,24)(20,28)(23,30)(25,31)(34,36)(35,42)(41,44)(43,48);(1,4,7,3)(2,5,9,10)(6,12,15,11)(8,13,16,14)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,5)(2,3)(4,9)(6,14)(7,10)(8,12)(11,16)(13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,6,7,15)(2,8,9,16)(3,11,4,12)(5,13,10,14)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,38,39,46)(34,41,36,44)(35,43,42,48)(37,45,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817236);