Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7504676 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,15)(5,10)(6,18)(8,17)(9,11)(12,28)(13,22)(14,23)(16,26)(19,24)(20,27)(21,29)(25,31)(30,32);(1,3,11,13,5,14,6,19)(2,7,18,24,10,23,9,22)(4,12,20,29,16,30,17,31)(8,25,26,32,27,21,15,28);(1,4,5,16)(2,8,10,27)(3,12)(6,20,11,17)(7,21)(9,26,18,15)(13,31)(14,30)(19,29)(22,28)(23,25)(24,32);(2,9)(6,11)(7,22)(8,26)(10,18)(13,19)(15,27)(17,20)(21,28)(23,24)(25,32)(29,31);(2,10)(7,23)(8,27)(9,18)(15,26)(21,25)(22,24)(28,32)Polycyclic presentation:
2^9|b^2=fg,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411872);