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 | #40648 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(20,28)(23,30)(26,32)(29,31)(33,34)(35,45)(36,42)(37,38)(39,47)(40,44)(41,48)(43,46);(2,7)(3,10)(9,13)(11,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(2,7)(3,10)(9,13)(11,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,36,40,38)(34,42,44,37)(35,43,48,47)(39,45,46,41);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,37,40,42)(34,36,44,38)(35,46,48,39)(41,43,45,47);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,38,40,36)(34,37,44,42)(35,47,48,43)(39,41,46,45);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,39,40,46)(34,43,44,47)(35,42,48,37)(36,45,38,41)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064941);