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 | #40670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(4,11)(12,16)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(33,34)(35,42)(36,46)(37,41)(38,39)(40,45)(43,44)(47,48);(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(19,29)(20,31)(24,28)(25,30)(34,39)(36,41)(40,42)(43,47);(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,35,38,45)(34,40,39,42)(36,43,41,47)(37,44,46,48);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,36)(34,37)(35,43)(38,41)(39,46)(40,44)(42,48)(45,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,37)(34,41)(35,44)(36,39)(38,46)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8352903);