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 | |
3773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(6,10)(11,15)(14,16)(17,18,23,20)(19,22,27,24)(21,25,29,28)(26,30,32,31)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,2)(3,10)(4,12)(5,8)(6,9)(7,13)(11,14)(15,16)(21,29)(25,28)(26,32)(30,31)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,19,23,27)(18,24,20,22)(21,26,29,32)(25,31,28,30)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,22,27,24)(21,28,29,25)(26,30,32,31)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,3,5,9)(2,6,8,10)(4,11,13,15)(7,14,12,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,22)(18,19)(20,27)(21,30)(23,24)(25,26)(28,32)(29,31)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664227);