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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,9,6,8)(4,13,15,5)(11,12,14,16)(17,18,22,26)(19,23,24,20)(21,29,30,25)(27,28,32,31)(36,44)(40,47)(42,48)(45,46);(1,3,7,6)(2,8,10,9)(4,11,15,14)(5,12,13,16)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,47)(45,48);(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,7,15)(2,5,10,13)(3,11,6,14)(8,12,9,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5)(2,4)(3,12)(6,16)(7,13)(8,11)(9,14)(10,15)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,34)(35,38)(36,40)(37,41)(39,43)(42,45)(44,47)(46,48);(1,6,7,3)(2,9,10,8)(4,14,15,11)(5,16,13,12)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044376);