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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4160 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)(3,6)(4,7)(5,10)(8,13)(9,11)(12,15)(14,16)(34,39)(35,41)(40,44)(43,46);(2,3)(7,9)(8,10)(14,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,34,38,39)(35,42,41,37)(36,44,45,40)(43,47,46,48);(2,3)(4,11)(5,13)(14,15)(18,22)(19,26)(20,29)(31,32)(35,41)(36,45)(37,42)(40,44);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,37)(36,46)(38,41)(39,42)(40,48)(43,45)(44,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36,38,45)(34,40,39,44)(35,43,41,46)(37,47,42,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37)(34,35)(36,47)(38,42)(39,41)(40,43)(44,46)(45,48)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965631);