Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23861 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,18,23,27)(19,24,20,29)(21,26,22,31)(25,28,32,30);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,23,20)(18,24,27,29)(21,30,22,28)(25,31,32,26);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=fgi,d^2=fgi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72257340, #72257454, #72263999, … |
| maximal normal subgroup of | #72257340, #72257454, #72263999, … |
| maximal quotient of | #72307776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7497311);