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 | |
3958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,13,11)(5,14,16,9)(6,10,12,15)(17,18,23,27)(19,28,30,24)(20,21,29,25)(22,26,32,31)(33,34,39,38)(35,40,45,44)(36,46,41,37)(42,48,47,43);(1,3)(2,8)(4,11)(5,6)(7,13)(9,15)(10,14)(12,16)(17,19,23,30)(18,24,27,28)(20,22,29,32)(21,31,25,26)(33,35,39,45)(34,40,38,44)(36,42,41,47)(37,43,46,48);(1,2,7,4)(3,11,13,8)(5,9,16,14)(6,10,12,15)(17,18,23,27)(19,24,30,28)(20,25,29,21)(22,26,32,31)(33,36)(34,37)(35,42)(38,46)(39,41)(40,43)(44,48)(45,47);(1,4,7,2)(3,11,13,8)(5,14,16,9)(6,10,12,15)(17,20)(18,25)(19,22)(21,27)(23,29)(24,26)(28,31)(30,32)(33,35)(34,40)(36,42)(37,43)(38,44)(39,45)(41,47)(46,48);(1,5)(2,9)(3,12)(4,14)(6,13)(7,16)(8,10)(11,15)(17,21,23,25)(18,20,27,29)(19,26,30,31)(22,24,32,28)(33,37,39,46)(34,41,38,36)(35,43,45,48)(40,47,44,42);(1,6,7,12)(2,10,4,15)(3,5,13,16)(8,14,11,9)(17,22)(18,26)(19,29)(20,30)(21,24)(23,32)(25,28)(27,31)(33,38,39,34)(35,44,45,40)(36,46,41,37)(42,48,47,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=i,e^2=gh,f^2=hi,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, 8412778);