Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24089 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61288 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
725 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)(3,13)(4,10)(5,18)(6,22)(7,12)(8,16)(9,20)(11,19)(14,26)(15,29)(17,28)(21,27)(23,24)(25,31)(30,32);(3,14,15,32)(4,5)(6,19)(8,24,25,28)(9,27)(10,18)(11,22)(13,30,29,26)(16,17,31,23)(20,21);(1,3,6,23,7,15,19,17)(2,8,11,30,12,25,22,26)(4,14,20,31,21,32,5,16)(9,28,10,29,18,24,27,13);(1,2)(3,8)(4,18)(5,10)(6,11)(7,12)(9,21)(13,31)(14,24)(15,25)(16,29)(17,26)(19,22)(20,27)(23,30)(28,32)Polycyclic presentation:
2^9|b^2=e,c^2=eh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72231593, #72231599, … |
| maximal normal subgroup of | #72213945, #72231593, #72231599, … |
| maximal quotient of | #72224942, #72237223, #72250461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407549);