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 | #47662 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34)(35,41)(36,42)(37,40)(38,39)(43,47)(44,48)(45,46);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(34,39)(35,42)(37,45)(44,47);(2,6)(3,9)(4,12)(14,16)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,35)(34,36)(37,43)(38,42)(39,41)(40,44)(45,48)(46,47);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,37,38,45)(34,40,39,46)(35,43,42,48)(36,44,41,47);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8957674);