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 | #36818 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4146 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)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(37,45)(40,44)(42,48)(46,47);(2,7)(4,8)(9,13)(12,15)(18,24)(19,20)(26,29)(27,28)(34,36)(39,41)(40,44)(46,47);(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,34,38,36)(35,39,43,41)(37,40,45,44)(42,46,48,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,13)(14,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,5,6,14)(2,9,7,13)(3,10,11,16)(4,12,8,15)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017705);