Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C2 | C2×#3109 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,14)(4,13)(5,12)(6,9)(7,20)(8,22)(10,15)(11,19)(16,30)(17,26)(18,25)(21,27)(23,29)(24,31)(28,32);(2,9)(4,18)(7,19)(10,24)(12,22)(15,31)(21,32)(27,28);(1,3)(2,10)(4,15)(5,16)(6,17)(7,21)(8,23)(9,24)(11,26)(12,27)(13,29)(14,20)(18,31)(19,32)(22,28)(25,30);(1,4,5,7,6,18,8,19)(2,11,22,25,9,20,12,13)(3,15,16,21,17,31,23,32)(10,26,28,30,24,14,27,29);(1,5,6,8)(2,12,9,22)(3,16,17,23)(4,7,18,19)(10,27,24,28)(11,13,20,25)(14,30,26,29)(15,21,31,32)Polycyclic presentation:
2^9|d^2=fh,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gh,h^a=hi,d^b=di,g^b=gi,f^c=fi,g^c=gh,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214311, #72214675, #72214826, … |
| maximal normal subgroup of | #72214311, #72214675, #72214826, … |
| maximal quotient of | #72224588, #72224792, C4×#33688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530343);