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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,4,8,7)(5,9,13,16)(11,12,15,14)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(1,3,6,8)(2,7,10,4)(5,11,13,15)(9,14,16,12)(18,24)(19,20)(26,29)(27,28)(34,40)(35,42)(41,44)(43,48);(5,13)(9,16)(11,15)(12,14)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,15)(14,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,4)(2,8)(3,10)(5,12)(6,7)(9,15)(11,16)(13,14)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,5,6,13)(2,9,10,16)(3,11,8,15)(4,12,7,14)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324392);