Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500001 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23960 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2932 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,13)(4,18)(5,22)(6,21)(8,26)(9,10)(11,24)(14,30)(15,23)(16,17)(19,25)(20,29)(27,31)(28,32);(1,2)(3,14,15,32)(4,10)(5,20)(6,24)(7,12)(8,16,27,19)(9,18)(11,21)(13,28,23,30)(17,26,25,31)(22,29);(1,3)(2,8)(4,19)(5,23)(6,25)(7,15)(9,14)(10,26)(11,30)(12,27)(13,18)(16,22)(17,21)(20,32)(24,28)(29,31);(1,2)(3,8)(4,20)(5,10)(6,11)(7,12)(9,22)(13,31)(14,16)(15,27)(17,28)(18,29)(19,32)(21,24)(23,26)(25,30)Polycyclic presentation:
2^9|b^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72231701, #72231712, … |
| maximal normal subgroup of | #72213937, #72231701, #72231712, … |
| maximal quotient of | #72231521, C2×#500001, #72239707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407197);