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 | #49682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3932 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)(3,5)(4,9)(6,7)(8,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,26,21,24)(20,30,29,25)(22,27,31,32)(34,38)(36,43)(39,44)(42,47);(2,7)(4,12)(5,8)(11,16)(17,19,23,21)(18,24,28,26)(20,31,29,22)(25,32,30,27)(35,41)(39,44)(42,47)(46,48);(2,7)(3,10)(4,12)(13,15)(17,20)(18,25)(19,22)(21,31)(23,29)(24,27)(26,32)(28,30)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,21,23,19)(18,26,28,24)(20,31,29,22)(25,32,30,27);(1,4,6,12)(2,9,7,14)(3,11,10,16)(5,13,8,15)(17,22,23,31)(18,27,28,32)(19,29,21,20)(24,30,26,25)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8717515);