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 | #39793 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3944 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)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,3,7,12)(2,6,8,11)(4,14,16,5)(9,10,13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(34,39)(35,42)(41,44)(43,48);(2,8)(6,11)(9,13)(10,15)(18,23)(21,31)(24,29)(25,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,4,7,16)(2,9,8,13)(3,5,12,14)(6,15,11,10)(17,19)(18,24)(20,26)(21,25)(22,28)(23,29)(27,30)(31,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5)(2,10)(3,4)(6,13)(7,14)(8,15)(9,11)(12,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,32)(24,31)(26,28)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48);(1,6,7,11)(2,3,8,12)(4,15,16,10)(5,13,14,9)(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,29,30,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=gh,f^2=gi,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, 8229445);