Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2067 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,25,21,24,28,22)(19,23,27,31,26,30,32,29);(9,11)(10,13)(12,15)(14,16)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(2,5)(6,8)(10,14)(11,15)(17,19,21,26)(18,23,24,30)(20,27,28,32)(22,29,25,31);(11,15)(13,16)(19,26)(23,30)(27,32)(29,31);(1,3,4,7)(2,6,5,8)(18,24)(22,25)(23,30)(29,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239658, #72251216, #72251240, … |
| maximal normal subgroup of | #72239658, #72251216, #72251240, … |
| maximal quotient of | #72239733, #72239769, #72308295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376026);