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 | #53525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3983 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(19,25)(23,26)(27,29)(31,32)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(2,6)(3,9)(8,13)(11,15)(17,18)(19,26)(20,28)(21,22)(23,25)(24,30)(27,31)(29,32)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(4,12)(8,13)(11,15)(14,16)(18,22)(23,26)(24,28)(31,32)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8323954);