Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394366 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,15)(5,20)(6,14)(8,11)(9,18)(10,22)(12,19)(13,23)(16,29)(17,25)(21,26)(24,32)(27,30)(28,31);(1,3,7,2,8,14,6,11)(4,9,18,15,19,25,17,12)(5,13,22,10,23,20,21,26)(16,24,31,28,32,29,30,27);(1,4)(2,9)(3,12)(5,16)(6,17)(7,18)(8,19)(10,24)(11,25)(13,27)(14,15)(20,28)(21,30)(22,31)(23,32)(26,29);(1,5)(2,10)(3,13)(4,16)(6,21)(7,22)(8,23)(9,24)(11,26)(12,27)(14,20)(15,28)(17,30)(18,31)(19,32)(25,29)Polycyclic presentation:
2^9|b^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72215968, #72215995, … |
| maximal normal subgroup of | #72215936, #72215968, #72215995, … |
| maximal quotient of | C2×#394367, #72252144, #72255745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301563);