Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500747 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24248 |
| 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 | |
1048 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)(5,23)(6,26)(7,12)(8,17)(9,21)(11,20)(14,30)(15,19)(16,31)(18,32)(22,25)(24,29)(27,28);(3,14,16,19)(4,5)(6,20)(8,28,29,32)(9,25)(10,23)(11,26)(13,15,31,30)(17,18,24,27)(21,22);(1,3)(2,8)(4,19)(5,24)(6,27)(7,16)(9,28)(10,13)(11,15)(12,29)(14,22)(17,21)(18,20)(23,32)(25,31)(26,30);(1,2)(3,15,16,30)(4,9)(5,25)(6,11)(7,12)(8,27,29,18)(10,21)(13,14,31,19)(17,32,24,28)(20,26)(22,23)Polycyclic presentation:
2^9|b^2=e,d^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=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72307951, C2×#500747 |
| maximal normal subgroup of | #72213945, #72307951, C2×#500747 |
| maximal quotient of | #72224936, #72239910, #72250656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407943);