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 | |
4096 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,28)(31,32)(33,34)(35,37)(36,40)(38,42)(39,41)(43,45)(44,46)(47,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,23)(20,24)(21,27)(22,25)(26,31)(28,29)(30,32)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(2,6)(7,13)(8,11)(14,16)(18,24)(19,25)(21,30)(28,31)(33,34,38,42)(35,39,41,37)(36,40,46,44)(43,47,48,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,31)(30,32)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,45)(47,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20)(18,25)(19,24)(21,28)(22,23)(26,32)(27,29)(30,31)(33,36,38,46)(34,40,42,44)(35,43,41,48)(37,45,39,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9385402);