Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(22,28)(29,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(2,8)(7,15)(9,13)(10,16)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)Polycyclic presentation:
2^9|d^2=i,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281720, #72284121, #72284184, … |
| maximal normal subgroup of | #72281720, #72284121, #72284184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253602);