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 | |
4165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,10,11)(5,8,13,12)(9,14,15,16)(19,25)(23,30)(26,28)(31,32)(33,34,39,37)(35,36,42,40)(38,45,46,41)(43,47,48,44);(2,4)(7,11)(8,12)(14,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(34,37)(35,42)(41,45)(43,48);(5,13)(8,12)(9,15)(14,16)(18,22)(23,30)(24,27)(31,32)(33,35,39,42)(34,40,37,36)(38,43,46,48)(41,47,45,44);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,15,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37,39,34)(35,36,42,40)(38,45,46,41)(43,44,48,47);(1,5,6,13)(2,8,4,12)(3,9,10,15)(7,14,11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,38,39,46)(34,41,37,45)(35,43,42,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832192);