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 | |
3971 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,16)(13,14)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(1,3,7,5)(2,8,9,11)(4,12,15,14)(6,13,10,16)(17,18,23,25)(19,26,20,22)(21,24,31,29)(27,32,28,30)(34,36)(35,41)(40,44)(43,47);(2,9)(6,10)(8,11)(13,16)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,16)(11,13)(17,20,23,19)(18,22,25,26)(21,28,31,27)(24,30,29,32)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,16,10,13)(17,21,23,31)(18,24,25,29)(19,27,20,28)(22,30,26,32)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,12,11,14)(17,22)(18,19)(20,25)(21,30)(23,26)(24,27)(28,29)(31,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095530);