Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437464 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2403 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6221 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)(3,11)(4,8)(5,17)(6,10)(7,13)(9,19)(12,27)(14,25)(15,23)(16,31)(18,21)(20,32)(22,29)(24,30)(26,28);(1,3)(2,7)(4,12)(5,18)(6,14)(8,22)(9,25)(10,21)(11,17)(13,19)(15,28)(16,32)(20,23)(24,26)(27,31)(29,30);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,24)(10,23)(11,26)(13,29)(14,28)(17,31)(18,32)(19,30)(21,22)(25,27);(5,19)(7,21)(9,17)(13,18)(16,30)(20,22)(24,31)(29,32);(4,15)(8,23)(12,28)(16,30)(20,22)(24,31)(26,27)(29,32)Polycyclic presentation:
2^9|f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,e^c=ei,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72233205, #72234100, #72235139, … |
| maximal normal subgroup of | #72233205, #72234100, #72235139, … |
| maximal quotient of | #72233137, #72233472, #72233807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344660);