Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24128 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2930 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(4,15)(7,21)(8,23)(11,26)(13,18)(16,30)(17,25)(29,31);(2,7)(4,16)(5,19)(8,22)(9,21)(10,25)(11,27)(13,29)(14,28)(15,30)(17,20)(18,31)(23,24)(26,32);(1,2)(3,11)(4,17)(5,20)(6,9)(7,16)(8,14)(10,19)(12,26)(13,22)(15,25)(18,24)(21,30)(23,28)(27,29)(31,32);(1,3)(2,8,9,23)(4,18)(5,14)(6,12)(7,22,21,24)(10,11,20,26)(13,15)(16,31)(17,32,25,27)(19,28)(29,30)Polycyclic presentation:
2^9|d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72250066, C2×#500471 |
| maximal normal subgroup of | #72213945, #72250066, C2×#500471 |
| maximal quotient of | #72224939, #72239904, #72250650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407667);