Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2403 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6084 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)(3,15)(4,19)(5,11)(6,12)(7,13)(8,14)(9,17)(10,20)(16,29)(18,28)(21,30)(22,31)(23,32)(24,26)(25,27);(1,3)(2,9)(4,5)(6,16)(7,17)(8,18)(10,11)(12,15)(13,28)(14,29)(19,32)(20,22)(21,24)(23,25)(26,27)(30,31);(1,4)(2,10)(3,5)(6,20)(7,25)(8,21)(9,26)(11,29)(12,30)(13,19)(14,27)(15,32)(16,22)(17,23)(18,24)(28,31);(1,5)(2,11)(3,4)(6,22)(7,23)(8,24)(9,27)(10,29)(12,31)(13,32)(14,26)(15,19)(16,20)(17,25)(18,21)(28,30);(1,6,8,7)(2,12,14,13)(3,16,18,17)(4,20,21,25)(5,22,24,23)(9,15,29,28)(10,30,27,19)(11,31,26,32)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72215958, #72215995, … |
| maximal normal subgroup of | #72215721, #72215958, #72215995, … |
| maximal quotient of | C4×#23882, #72235210, C2×#6437575, … |
| automorphism group of | #2165 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344771);