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 | #47047 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4003 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(2,7)(3,9)(10,13)(11,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(34,36)(35,41)(40,44)(43,47);(2,7)(4,8)(10,13)(12,15)(21,29)(25,32)(27,28)(30,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027103);