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 | |
4231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,31)(27,28)(30,32)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,48,44,42);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19)(18,23)(20,26)(21,27)(22,24)(25,32)(28,29)(30,31)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,42,46,48);(2,6)(7,14)(8,11)(15,16)(17,19,22,24)(18,20,26,23)(21,30,31,27)(25,32,29,28)(33,36,39,47)(34,41,37,46)(35,38,45,44)(40,48,43,42);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,42)(43,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20)(18,24)(19,26)(21,28)(22,23)(25,30)(27,29)(31,32)(33,37,39,34)(35,40,45,43)(36,46,47,41)(38,48,44,42);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21,22,31)(18,25,26,29)(19,27,24,30)(20,28,23,32)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,46,43,41)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388638);