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 | |
4027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,11)(3,8,12,7)(4,9,15,14)(5,10,16,13)(17,18)(19,25)(20,24)(21,32)(22,26)(23,27)(28,31)(29,30)(36,44)(40,45)(43,48)(46,47);(3,12)(4,15)(7,8)(9,14)(17,19,23,20)(18,24,27,25)(21,30,26,28)(22,29,32,31)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,46)(47,48);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,16)(10,15)(17,19,23,20)(18,25,27,24)(21,28,26,30)(22,29,32,31)(33,35)(34,37)(36,43)(38,41)(39,42)(40,46)(44,48)(45,47);(1,3)(2,8)(4,13)(5,9)(6,12)(7,11)(10,15)(14,16)(17,20,23,19)(18,25,27,24)(21,30,26,28)(22,29,32,31)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,48)(45,46);(1,4)(2,9)(3,13)(5,8)(6,15)(7,16)(10,12)(11,14)(17,21,23,26)(18,22,27,32)(19,28,20,30)(24,31,25,29)(33,36,38,44)(34,40,42,45)(35,43,41,48)(37,46,39,47);(1,5)(2,10)(3,14)(4,8)(6,16)(7,15)(9,12)(11,13)(17,22)(18,26)(19,29)(20,31)(21,27)(23,32)(24,28)(25,30)(33,37,38,39)(34,41,42,35)(36,46,44,47)(40,48,45,43)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413497);