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 | #54014 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3911 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,27)(20,30)(21,24)(22,26)(23,25)(28,31)(29,32)(33,34,39,44)(35,43,46,40)(36,41,47,48)(37,42,45,38);(3,9)(7,13)(10,15)(14,16)(17,19,24,23)(18,25,21,27)(20,29,22,28)(26,32,30,31)(33,35,39,46)(34,40,44,43)(36,45,47,37)(38,41,42,48);(2,6)(7,13)(8,11)(14,16)(17,20,24,22)(18,26,21,30)(19,28,23,29)(25,31,27,32)(33,36,39,47)(34,41,44,48)(35,45,46,37)(38,40,42,43);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,21,24,18)(19,27,23,25)(20,26,22,30)(28,31,29,32)(33,37)(34,42)(35,36)(38,44)(39,45)(40,48)(41,43)(46,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,22,24,20)(18,26,21,30)(19,29,23,28)(25,31,27,32)(33,35)(34,43)(36,45)(37,47)(38,41)(39,46)(40,44)(42,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,23,24,19)(18,27,21,25)(20,29,22,28)(26,32,30,31)(33,38,39,42)(34,37,44,45)(35,41,46,48)(36,43,47,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388658);