Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2425 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,26)(21,23)(22,25)(27,32)(28,30)(29,31);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,22)(24,31)(28,29)(30,32);(1,4,5,8)(2,6,3,7)(10,14)(13,16)(17,19,21,28)(18,24,23,30)(20,27,22,29)(25,31,26,32);(2,3)(6,7)(10,14)(13,16)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,31,30,32);(17,20,21,22)(18,26,23,25)(19,27,28,29)(24,32,30,31)Polycyclic presentation:
2^9|b^2=fgi,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281793, #72281815, #72282106, … |
| maximal normal subgroup of | #72281793, #72281815, #72282106, … |
| maximal quotient of | C2×#7504687, #72293234 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411883);