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 | |
4054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,11,12,6)(4,13,16,9)(5,15,14,10)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,34,39,42)(35,40,43,38)(36,37,41,47)(44,45,48,46);(1,3,7,12)(2,6,8,11)(4,14,16,5)(9,10,13,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,35,39,43)(34,40,42,38)(36,44,41,48)(37,45,47,46);(2,8)(6,11)(9,13)(10,15)(18,22)(23,25)(24,28)(31,32)(35,43)(38,40)(44,48)(45,46);(1,4,7,16)(2,9,8,13)(3,5,12,14)(6,15,11,10)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,5)(2,10)(3,4)(6,13)(7,14)(8,15)(9,11)(12,16)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,37)(34,41)(35,45)(36,42)(38,44)(39,47)(40,48)(43,46);(1,6,7,11)(2,3,8,12)(4,15,16,10)(5,13,14,9)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,38)(34,35)(36,46)(37,44)(39,40)(41,45)(42,43)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8192393);