Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #466 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,9,6,3,10,8)(4,7,12,15,13,11,16,14)(17,18,21,25,23,26,22,38)(19,28,31,46,29,47,32,44)(20,33,36,43,35,45,37,27)(24,39,41,30,42,34,40,48);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,14)(15,16)(17,19)(18,24)(20,34)(21,32)(22,31)(23,29)(25,40)(26,42)(27,44)(28,45)(30,36)(33,47)(35,39)(37,48)(38,41)(43,46);(1,4,10,16,6,13,5,12)(2,7,8,14,3,11,9,15)(18,25,26,38)(20,35)(24,40,42,41)(27,45,43,33)(28,46,47,44)(30,48)(34,39)(36,37);(2,3)(7,11)(8,9)(14,15)(18,26)(19,29)(25,38)(27,43)(30,48)(31,32)(33,45)(34,39)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=dh,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286498, #72286506, #72286537, … |
| maximal normal subgroup of | #72286498, #72286506, #72286537, … |
| maximal quotient of | #72343640 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391183);