Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147842 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10995 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #60753 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,15,25)(4,16,18,8)(5,20)(6,10)(7,13)(9,21)(12,27,14,30)(17,23)(19,26,32,31)(22,29)(24,28);(1,3)(2,7)(4,17)(5,12)(6,15)(8,25)(9,22)(10,23)(11,16)(13,18)(14,21)(19,29)(20,24)(26,27)(28,32)(30,31);(3,12)(5,21)(7,22)(8,16)(9,20)(11,27)(13,28)(14,15)(17,29)(19,32)(23,24)(25,30)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72240505, #72258095, #72258266, … |
| maximal normal subgroup of | #72240505, #72258095, #72258266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55038);