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 | |
4122 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)(3,6)(4,7)(5,10)(8,13)(9,11)(12,15)(14,16)(33,34,37,41)(35,39,42,38)(36,40,45,48)(43,47,44,46);(2,3)(7,9)(8,10)(14,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(35,42)(38,39)(43,44)(46,47);(2,3)(4,11)(5,13)(14,15)(18,22)(19,26)(20,29)(31,32)(33,35,37,42)(34,38,41,39)(36,44,45,43)(40,47,48,46);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,39)(36,44)(37,42)(38,41)(40,46)(43,45)(47,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36,37,45)(34,40,41,48)(35,43,42,44)(38,46,39,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966152);