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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(18,22)(23,26)(24,28)(30,32)(33,34,39,42)(35,40,36,38)(37,41,46,48)(43,47,44,45);(2,7)(3,11)(5,14)(12,15)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30)(35,36)(38,40)(43,44)(45,47);(2,7)(4,9)(8,10)(12,15)(19,25)(23,26)(27,31)(30,32)(33,35,39,36)(34,40,42,38)(37,44,46,43)(41,45,48,47);(1,3,6,11)(2,8,7,10)(4,12,9,15)(5,13,14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,39,35)(34,38,42,40)(37,43,46,44)(41,47,48,45);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38)(34,35)(36,42)(37,45)(39,40)(41,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10270865);