Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,12)(4,9)(8,21)(10,19)(11,20)(13,28)(14,27)(15,25)(16,29)(18,23)(22,26)(24,32)(30,31);(1,2,4,7,6,11,9,20)(3,8,13,18,14,24,22,31)(5,16,15,29,17,10,25,19)(12,23,26,21,27,30,28,32);(1,3)(2,8)(4,13)(5,12)(6,14)(7,18)(9,22)(10,23)(11,24)(15,26)(16,30)(17,27)(19,21)(20,31)(25,28)(29,32);(1,2,9,20,6,11,4,7)(3,8,22,31,14,24,13,18)(5,10,25,29,17,16,15,19)(12,23,28,32,27,30,26,21)Polycyclic presentation:
2^9|b^2=e,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72236829, #72236853, #72236890, … |
| maximal normal subgroup of | #72236890, #72236997, #72237523, … |
| maximal quotient of | #72311949, #72312172, #72312358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408001);