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 |
| prime radical | #50115 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34)(35,40)(36,38)(37,42)(39,41)(43,47)(44,46)(45,48);(2,6)(3,9)(8,13)(11,15)(18,23)(19,26)(24,29)(27,32)(33,35)(34,40)(36,37)(38,42)(39,44)(41,46)(43,45)(47,48);(3,9)(4,12)(7,10)(8,13)(18,23)(20,25)(21,30)(27,32)(34,41)(35,44)(37,45)(38,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,36,39,43)(34,38,41,47)(35,45,44,37)(40,48,46,42);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,37,39,45)(34,42,41,48)(35,36,44,43)(38,46,47,40);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38)(34,43)(35,42)(36,41)(37,46)(39,47)(40,45)(44,48)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947901);