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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3985 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)(5,11)(6,13)(7,9)(12,16)(14,15)(33,34,39,43)(35,36,44,40)(37,41,47,46)(38,45,48,42);(1,3,7,5)(2,8,9,11)(4,12,14,13)(6,10,16,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35)(34,40)(36,43)(37,38)(39,44)(41,42)(45,46)(47,48);(2,9)(3,5)(4,14)(6,16)(18,22)(19,26)(20,29)(31,32)(33,36,39,40)(34,35,43,44)(37,42,47,45)(38,41,48,46);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,36)(34,35)(37,42)(38,46)(39,40)(41,48)(43,44)(45,47);(1,5,7,3)(2,11,9,8)(4,13,14,12)(6,15,16,10)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,37)(34,41)(35,38)(36,45)(39,47)(40,42)(43,46)(44,48);(1,6)(2,12)(3,10)(4,11)(5,15)(7,16)(8,14)(9,13)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38)(34,42)(35,37)(36,46)(39,48)(40,41)(43,45)(44,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,e^2=gh,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, 8966384);