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 | |
4290 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)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48);(2,6)(7,13)(8,11)(14,16)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,35)(34,39)(36,42)(37,41)(38,45)(40,43)(44,46)(47,48);(3,9)(7,13)(10,15)(14,16)(18,23)(19,26)(24,29)(27,32)(33,35,38,45)(34,39,43,40)(36,44,46,42)(37,41,47,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,35)(34,40)(36,44)(37,41)(38,45)(39,43)(42,46)(47,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36)(34,41)(35,44)(37,40)(38,46)(39,47)(42,45)(43,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,44)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9877300);