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 | |
4174 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(33,34,38,40)(35,39,44,43)(36,41,45,46)(37,42,47,48);(2,7)(4,15)(8,11)(10,13)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(33,35)(34,39)(36,42)(37,46)(38,44)(40,43)(41,47)(45,48);(2,7)(3,12)(4,15)(5,16)(18,24)(20,28)(23,30)(25,31)(34,40)(35,44)(36,45)(37,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,36,38,45)(34,41,40,46)(35,42,44,48)(37,39,47,43);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37)(34,42)(35,41)(36,43)(38,47)(39,45)(40,48)(44,46);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,35)(34,43)(36,42)(37,41)(38,44)(39,40)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9414097);