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 | #2537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,11,14,8)(4,15,13,6)(5,16,12,9)(19,27)(20,29)(23,30)(24,32)(33,34,38,40)(35,41,42,36)(37,39,46,48)(43,45,47,44);(1,3)(2,8)(4,12)(5,13)(6,9)(7,14)(10,11)(15,16)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,28,31)(35,42)(36,41)(43,47)(44,45);(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,32)(26,30)(33,35)(34,36)(37,43)(38,42)(39,44)(40,41)(45,48)(46,47);(1,4,7,13)(2,6,10,15)(3,12,14,5)(8,9,11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36)(34,35)(37,45)(38,41)(39,47)(40,42)(43,48)(44,46);(1,5,7,12)(2,9,10,16)(3,13,14,4)(6,11,15,8)(17,20,22,29)(18,24,26,32)(19,28,27,21)(23,31,30,25);(1,6)(2,4)(3,9)(5,11)(7,15)(8,12)(10,13)(14,16)(17,21,22,28)(18,25,26,31)(19,29,27,20)(23,32,30,24)(33,37)(34,39)(35,43)(36,44)(38,46)(40,48)(41,45)(42,47)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=g,e^2=ghi,f^2=hi,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, 8370858);