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 |
| prime radical | #55436 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3993 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)(3,8)(4,10)(5,12)(6,11)(7,9)(13,16)(14,15)(33,34,38,43)(35,42,45,39)(36,41,37,40)(44,47,46,48);(1,3)(2,8)(4,6)(5,9)(7,12)(10,11)(13,14)(15,16)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(33,35)(34,39)(36,44)(37,46)(38,45)(40,47)(41,48)(42,43);(2,9)(3,12)(6,13)(10,15)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,36,38,37)(34,40,43,41)(35,44,45,46)(39,47,42,48);(1,4)(2,10)(3,13)(5,11)(6,12)(7,14)(8,16)(9,15)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,34)(35,42)(36,40)(37,41)(38,43)(39,45)(44,48)(46,47);(1,5)(2,3)(4,11)(6,15)(7,8)(9,12)(10,13)(14,16)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,37,38,36)(34,41,43,40)(35,44,45,46)(39,47,42,48);(1,6,7,13)(2,11,9,16)(3,4,12,14)(5,15,8,10)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,35)(34,42)(36,46)(37,44)(38,45)(39,43)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=ghi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8178392);