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 | #58436 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3893 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)(3,8,13,15)(4,5,10,9)(11,16,14,12)(19,25)(23,26)(27,29)(31,32)(33,34,38,42)(35,39,45,48)(36,37,41,40)(43,44,47,46);(2,7)(4,10)(8,15)(11,14)(17,18)(19,26)(20,28)(21,22)(23,25)(24,30)(27,31)(29,32)(33,35,38,45)(34,39,42,48)(36,43,41,47)(37,44,40,46);(2,7)(5,9)(8,15)(12,16)(18,22)(23,26)(24,28)(31,32)(33,36,38,41)(34,40,42,37)(35,43,45,47)(39,46,48,44);(1,3,6,13)(2,8,7,15)(4,14,10,11)(5,12,9,16)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,35)(34,39)(36,43)(37,44)(38,45)(40,46)(41,47)(42,48);(1,4,6,10)(2,9,7,5)(3,11,13,14)(8,12,15,16)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,9)(2,10,7,4)(3,12,13,16)(8,11,15,14)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,37,38,40)(34,41,42,36)(35,44,45,46)(39,47,48,43)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=gi,e^2=gi,f^2=h,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, 8324024);