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 | #35473 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4059 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)(3,7)(4,6)(5,8)(9,11)(10,14)(12,13)(15,16)(19,25)(20,28)(23,26)(24,29)(33,34,39,43)(35,42,38,40)(36,45,37,41)(44,47,46,48);(2,4)(3,9)(8,12)(10,15)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,35,39,38)(34,40,43,42)(36,44,37,46)(41,47,45,48);(5,13)(8,12)(10,15)(14,16)(18,22)(23,26)(24,29)(31,32)(33,36,39,37)(34,41,43,45)(35,44,38,46)(40,47,42,48);(1,3,6,9)(2,7,4,11)(5,10,13,15)(8,14,12,16)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,37,39,36)(34,41,43,45)(35,44,38,46)(40,48,42,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,38,39,35)(34,42,43,40)(36,46,37,44)(41,48,45,47);(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)(33,34)(35,40)(36,45)(37,41)(38,42)(39,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8839959);