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 | #38849 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3949 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,15)(12,16)(17,18,23,26)(19,24,20,22)(21,25,30,32)(27,31,28,29)(33,34,39,43)(35,38,45,47)(36,42,46,40)(37,41,44,48);(2,7)(3,10)(5,14)(12,15)(19,20)(22,24)(27,28)(29,31)(33,35,39,45)(34,38,43,47)(36,37,46,44)(40,48,42,41);(2,7)(4,8)(9,13)(12,15)(17,19,23,20)(18,24,26,22)(21,28,30,27)(25,29,32,31)(33,36,39,46)(34,40,43,42)(35,44,45,37)(38,41,47,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,20,23,19)(18,22,26,24)(21,27,30,28)(25,31,32,29)(33,37)(34,41)(35,36)(38,42)(39,44)(40,47)(43,48)(45,46);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,36)(34,42)(35,37)(38,41)(39,46)(40,43)(44,45)(47,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(28,32)(30,31)(33,38)(34,35)(36,41)(37,42)(39,47)(40,44)(43,45)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10270905);