Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454434 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16252 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1890 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)(3,14)(4,9)(5,10)(6,12)(7,13)(8,16)(11,21)(15,28)(17,24)(18,22)(19,26)(20,27)(23,29)(25,30)(31,32);(1,3,6,16,7,17,21,29)(2,8,11,14,13,23,12,24)(4,18,19,30,20,31,5,15)(9,25,10,22,27,28,26,32);(1,4)(2,9)(3,15)(5,6)(7,20)(8,22)(10,12)(11,26)(13,27)(14,25)(16,31)(17,30)(18,29)(19,21)(23,32)(24,28);(1,5,21,20,7,19,6,4)(2,10,12,9,13,26,11,27)(3,15,29,31,17,30,16,18)(8,22,24,25,23,32,14,28)Polycyclic presentation:
2^9|b^2=e,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72215995, #72216037, … |
| maximal normal subgroup of | #72215958, #72215995, #72216037, … |
| maximal quotient of | C2×#454435, #72252148, #72265248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361631);