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 | |
4177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,22)(20,23)(21,25)(24,26)(27,30)(28,31)(29,32)(33,34,38,41)(35,39,43,36)(37,45,46,40)(42,44,48,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,6)(7,9)(8,13)(14,15)(18,20)(19,26)(25,28)(27,29)(33,35,38,43)(34,36,41,39)(37,42,46,48)(40,44,45,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,18)(19,24,26,22)(21,28,31,25)(27,32,29,30)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,30,24,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,24)(18,26,20,19)(21,30,31,32)(25,29,28,27)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000791);