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 | |
4338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,11,15)(4,9,13,12)(5,10,16,14)(17,18,22,23)(19,24,27,20)(21,29,30,25)(26,28,32,31)(33,34)(35,40)(36,39)(37,41)(38,44)(42,46)(43,45)(47,48);(2,7)(4,13)(8,15)(10,14)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,31)(30,32)(33,35)(34,40)(36,37)(38,43)(39,41)(42,47)(44,45)(46,48);(3,11)(5,16)(8,15)(10,14)(18,23)(20,24)(21,30)(26,32)(34,36)(35,41)(38,46)(45,47);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,20)(18,19)(21,28)(22,24)(23,27)(25,26)(29,32)(30,31)(33,36,39,34)(35,37,41,40)(38,44,46,42)(43,45,48,47);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,29)(30,32)(33,37)(34,41)(35,36)(38,45)(39,40)(42,48)(43,44)(46,47);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834406);