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 | #55024 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4018 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)(3,4)(5,15)(6,11)(7,9)(8,13)(10,12)(14,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34,39,37)(35,43,46,40)(36,41,45,47)(38,42,44,48);(1,3,7,13)(2,8,9,4)(5,6,12,16)(10,11,15,14)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,42)(47,48);(2,9)(4,8)(10,15)(11,14)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,42,43,48);(1,4,7,8)(2,3,9,13)(5,14,12,11)(6,15,16,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36)(34,41)(35,44)(37,47)(38,46)(39,45)(40,42)(43,48);(1,5)(2,10)(3,6)(4,14)(7,12)(8,11)(9,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,37,39,34)(35,43,46,40)(36,47,45,41)(38,42,44,48);(1,6)(2,11)(3,12)(4,10)(5,13)(7,16)(8,15)(9,14)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,41,43,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8648781);