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 | |
4050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,24,27)(19,28,31,25)(20,32,22,23)(21,29,30,26)(33,34)(35,43)(36,41)(37,42)(38,39)(40,44)(45,48)(46,47);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,24,31)(18,25,27,28)(20,29,22,26)(21,23,30,32)(35,44)(37,45)(40,43)(42,48);(2,4)(3,10)(9,12)(11,16)(17,20,24,22)(18,23,27,32)(19,29,31,26)(21,25,30,28)(33,35,39,44)(34,40,38,43)(36,45,47,37)(41,42,46,48);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,21,24,30)(18,26,27,29)(19,23,31,32)(20,28,22,25)(33,36,39,47)(34,41,38,46)(35,37,44,45)(40,48,43,42);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,22,24,20)(18,23,27,32)(19,29,31,26)(21,28,30,25)(33,37)(34,42)(35,36)(38,48)(39,45)(40,46)(41,43)(44,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,23)(18,22)(19,30)(20,27)(21,31)(24,32)(25,26)(28,29)(33,38,39,34)(35,43,44,40)(36,46,47,41)(37,42,45,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gi,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073358);