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 | |
4019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18,23,27)(19,28,30,24)(20,22,26,31)(21,25,29,32)(33,34,39,43)(35,42,38,40)(36,41,47,37)(44,48,46,45);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19)(18,24)(20,29)(21,26)(22,25)(23,30)(27,28)(31,32)(33,35,39,38)(34,40,43,42)(36,44,47,46)(37,48,41,45);(2,7)(3,10)(8,13)(11,16)(33,36,39,47)(34,37,43,41)(35,44,38,46)(40,48,42,45);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,23,26)(18,22,27,31)(19,29,30,21)(24,25,28,32)(33,36)(34,41)(35,44)(37,43)(38,46)(39,47)(40,45)(42,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,23,29)(18,25,27,32)(19,26,30,20)(22,28,31,24)(33,37,39,41)(34,36,43,47)(35,45,38,48)(40,46,42,44);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,15)(10,16)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,29)(30,32)(33,38,39,35)(34,42,43,40)(36,46,47,44)(37,45,41,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9633650);