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 | |
4097 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)(12,14)(13,15)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34)(35,37)(36,40)(38,41)(39,43)(42,45)(44,48)(46,47);(2,6)(7,9)(8,12)(15,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(2,6)(7,9)(8,12)(15,16)(19,26)(22,23)(27,32)(30,31)(36,44)(40,48)(42,46)(45,47);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,37,38,39)(34,35,41,43)(36,45,44,47)(40,42,48,46);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,35)(34,37)(36,46)(38,43)(39,41)(40,47)(42,44)(45,48)Polycyclic presentation:
2^9|e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635847);