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 | |
4053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,12,4,8)(5,10,11,6)(13,14,15,16)(19,25)(23,30)(26,28)(31,32)(33,34,38,39)(35,37,43,40)(36,41,46,47)(42,45,44,48);(1,3,7,4)(2,8,9,12)(5,13,11,15)(6,14,10,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,34,38,39)(35,40,43,37)(36,41,46,47)(42,48,44,45);(2,9)(6,10)(8,12)(14,16)(18,22)(23,30)(24,27)(31,32)(34,39)(37,40)(41,47)(45,48);(1,4,7,3)(2,8,9,12)(5,13,11,15)(6,16,10,14)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,35)(34,40)(36,44)(37,39)(38,43)(41,45)(42,46)(47,48);(1,5,7,11)(2,10,9,6)(3,13,4,15)(8,16,12,14)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,36)(34,41)(35,42)(37,45)(38,46)(39,47)(40,48)(43,44);(1,6,7,10)(2,11,9,5)(3,14,4,16)(8,15,12,13)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,37,38,40)(34,35,39,43)(36,45,46,48)(41,42,47,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=g,f^2=h,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, 8833156);