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 | |
4096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,6,4,11)(5,9,14,16)(10,15,12,13)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(33,34,38,41)(35,39,43,36)(37,45,47,40)(42,44,48,46);(1,3)(2,6)(4,7)(5,10)(8,11)(9,15)(12,14)(13,16)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,47)(46,48);(2,8)(3,4)(9,16)(10,12)(18,24)(20,28)(23,30)(25,31)(33,35,38,43)(34,36,41,39)(37,42,47,48)(40,44,45,46);(1,4,7,3)(2,6,8,11)(5,12,14,10)(9,15,16,13)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,5)(2,9)(3,10)(4,12)(6,15)(7,14)(8,16)(11,13)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36,38,39)(34,35,41,43)(37,46,47,44)(40,48,45,42);(1,6)(2,3)(4,8)(5,13)(7,11)(9,12)(10,16)(14,15)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,37)(34,40)(35,42)(36,44)(38,47)(39,46)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9854194);