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 | |
4094 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,13)(14,15)(17,18)(19,26)(20,29)(21,23)(22,25)(24,30)(27,31)(28,32)(33,34,39,43)(35,40,44,48)(36,38,42,47)(37,46,45,41);(1,3,7,4)(2,8,11,9)(5,14,16,12)(6,13,10,15)(17,19,22,21)(18,23,25,26)(20,27,30,28)(24,31,29,32)(35,44)(37,45)(40,48)(41,46);(1,3,7,4)(2,9,11,8)(5,12,16,14)(6,13,10,15)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,29)(26,32)(33,35,39,44)(34,40,43,48)(36,45,42,37)(38,41,47,46);(1,4,7,3)(2,9,11,8)(5,14,16,12)(6,13,10,15)(17,20)(18,24)(19,28)(21,27)(22,30)(23,32)(25,29)(26,31)(33,36)(34,38)(35,45)(37,44)(39,42)(40,41)(43,47)(46,48);(1,5)(2,10)(3,12)(4,14)(6,11)(7,16)(8,13)(9,15)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,37,39,45)(34,41,43,46)(35,36,44,42)(38,40,47,48);(1,6,7,10)(2,5,11,16)(3,13,4,15)(8,14,9,12)(17,21,22,19)(18,23,25,26)(20,28,30,27)(24,31,29,32)(33,38)(34,42)(35,41)(36,43)(37,40)(39,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400128);