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 | #50843 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,11,12,4)(5,9,15,16)(6,14,13,10)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3,7,12)(2,4,8,11)(5,13,15,6)(9,14,16,10)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(2,8)(4,11)(9,16)(10,14)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4,7,11)(2,3,8,12)(5,14,15,10)(6,9,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,5,7,15)(2,9,8,16)(3,6,12,13)(4,10,11,14)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,6)(2,10)(3,5)(4,9)(7,13)(8,14)(11,16)(12,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091092);