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 |
| prime radical | #52682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3875 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)(17,18,23,20)(19,24,29,27)(21,25,28,30)(22,26,32,31)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(2,7)(3,10)(8,13)(11,16)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,23,18)(19,27,29,24)(21,25,28,30)(22,26,32,31)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,23,28)(18,25,20,30)(19,22,29,32)(24,26,27,31)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,15)(10,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=hi,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^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575477);