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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,12,13,8)(10,14,16,15)(17,18,23,27)(19,24,29,32)(20,21,25,30)(22,31,28,26)(33,34)(35,44)(36,37)(38,45)(39,43)(40,48)(41,47)(42,46);(2,6)(7,9)(8,12)(14,15)(17,19,23,29)(18,24,27,32)(20,28,25,22)(21,26,30,31)(33,35)(34,40)(36,45)(37,46)(38,47)(39,48)(41,42)(43,44);(2,6)(7,9)(8,12)(14,15)(17,20,23,25)(18,21,27,30)(19,22,29,28)(24,31,32,26)(33,36,39,41)(34,37,43,47)(35,45,48,42)(38,40,46,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,21)(19,28)(22,29)(23,25)(24,26)(27,30)(31,32)(33,37,39,47)(34,41,43,36)(35,46,48,38)(40,42,44,45);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21)(18,25)(19,26)(20,27)(22,24)(23,30)(28,32)(29,31)(33,34)(35,40)(36,37)(38,42)(39,43)(41,47)(44,48)(45,46);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22,23,28)(18,26,27,31)(19,20,29,25)(21,24,30,32)(33,38,39,46)(34,42,43,45)(35,47,48,37)(36,40,41,44)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081579);