Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1406 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(3,10)(6,9)(11,16)(13,15);(2,7)(6,9)(8,14)(13,15);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(18,22)(23,28)(24,29)(30,32)Polycyclic presentation:
2^9|a^2=g,e^2=f,g^2=h,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf176 conjugacy classes — view the full character table.
| maximal subgroup of | #72287655, #72287662, #72287677, … |
| maximal normal subgroup of | #72287655, #72287662, #72287677, … |
| maximal quotient of | #72343785 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529945);