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 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,10,13,7)(4,5,14,9)(11,15,16,12)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,39,44)(35,40,46,37)(36,42,48,41)(38,47,45,43);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,15)(14,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35,39,46)(34,40,44,37)(36,45,48,38)(41,47,42,43);(2,8)(4,14)(7,10)(11,16)(18,20)(19,26)(25,28)(27,32)(33,36,39,48)(34,41,44,42)(35,38,46,45)(37,47,40,43);(1,4)(2,9)(3,11)(5,8)(6,14)(7,15)(10,12)(13,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36)(34,42)(35,45)(37,47)(38,46)(39,48)(40,43)(41,44);(1,3)(2,10)(4,11)(5,15)(6,13)(7,8)(9,12)(14,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37)(34,35)(36,47)(38,41)(39,40)(42,45)(43,48)(44,46);(1,5,6,9)(2,4,8,14)(3,12,13,15)(7,11,10,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,38,39,45)(34,43,44,47)(35,36,46,48)(37,42,40,41)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9722502);