Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6451783 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2404 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,8)(5,6)(9,13)(10,23)(11,12)(16,17)(18,26)(19,21)(22,25)(24,27)(28,32)(30,31);(1,2)(3,13)(4,18)(5,12)(6,8)(7,11)(9,17)(10,19)(14,28)(15,22)(16,25)(20,23)(21,26)(24,29)(27,30)(31,32);(1,3)(2,9,11,25)(4,14)(5,16)(6,17)(7,15)(8,22,12,13)(10,24,26,28)(18,27,23,32)(19,30)(20,29)(21,31);(1,4)(2,10,11,26)(3,14)(5,21)(6,19)(7,20)(8,23,12,18)(9,24,25,28)(13,27,22,32)(15,29)(16,31)(17,30);(3,15)(9,25)(13,22)(14,29)(16,17)(24,28)(27,32)(30,31)Polycyclic presentation:
2^9|c^2=g,d^2=g,f^2=i,h^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,h^b=hi,e^c=ei,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233777, #72233778, … |
| maximal normal subgroup of | #72233772, #72233777, #72233778, … |
| maximal quotient of | #72256015, #72262291, #72318245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358979);