Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437950 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2410 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,13,8,12,7,14)(3,15,16,26,18,27,17,9)(4,19,21,29,23,32,22,10)(5,11,20,28,24,30,25,31);(1,3)(2,9)(4,20)(5,21)(6,17)(7,16)(8,18)(10,28)(11,19)(12,26)(13,27)(14,15)(22,24)(23,25)(29,31)(30,32);(1,4)(2,10)(3,5)(6,21)(7,22)(8,23)(9,11)(12,29)(13,19)(14,32)(15,28)(16,20)(17,25)(18,24)(26,30)(27,31);(1,5)(2,11)(3,4)(6,20)(7,25)(8,24)(9,10)(12,30)(13,28)(14,31)(15,19)(16,21)(17,22)(18,23)(26,29)(27,32);(2,12)(9,26)(10,29)(11,30)(13,14)(15,27)(19,32)(28,31)Polycyclic presentation:
2^9|a^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,g^c=gi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72214082, #72215721, … |
| maximal normal subgroup of | #72213913, #72214082, #72215721, … |
| maximal quotient of | #72239611, #72239618, #72250771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345146);