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 | |
3985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,12,14,8)(4,13,15,9)(5,6,10,16)(17,18,23,26)(19,24,27,22)(20,21,25,32)(28,29,30,31)(33,34,39,37)(35,42,41,38)(36,45,47,40)(43,48,44,46);(1,3,7,14)(2,8,11,12)(4,5,15,10)(6,13,16,9)(17,19,23,27)(18,24,26,22)(20,30,25,28)(21,31,32,29)(33,35)(34,38)(36,43)(37,42)(39,41)(40,48)(44,47)(45,46);(4,15)(5,10)(6,16)(9,13)(19,27)(20,25)(21,32)(22,24)(33,36,39,47)(34,40,37,45)(35,43,41,44)(38,48,42,46);(1,4,7,15)(2,9,11,13)(3,10,14,5)(6,8,16,12)(17,20)(18,21)(19,28)(22,31)(23,25)(24,29)(26,32)(27,30)(33,36)(34,40)(35,44)(37,45)(38,46)(39,47)(41,43)(42,48);(1,5,7,10)(2,6,11,16)(3,4,14,15)(8,9,12,13)(17,21)(18,25)(19,29)(20,26)(22,28)(23,32)(24,30)(27,31)(33,37,39,34)(35,42,41,38)(36,40,47,45)(43,48,44,46);(1,6)(2,10)(3,13)(4,8)(5,11)(7,16)(9,14)(12,15)(17,22)(18,19)(20,31)(21,28)(23,24)(25,29)(26,27)(30,32)(33,38,39,42)(34,41,37,35)(36,46,47,48)(40,43,45,44)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969364);