Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623148 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#3109 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,12)(4,16)(5,17)(7,20)(8,24)(10,26)(11,25)(13,19)(14,30)(15,31)(18,32)(21,23)(22,29)(27,28);(1,2)(3,7)(4,8)(5,11)(6,9)(10,18)(12,29)(13,21)(14,22)(15,23)(16,24)(17,25)(19,28)(20,30)(26,32)(27,31);(1,3)(2,7)(4,13)(5,19)(6,14)(8,21)(9,22)(10,27)(11,28)(12,16)(15,17)(18,31)(20,26)(23,25)(24,29)(30,32);(1,4,5,16,6,18,17,32)(2,8,11,24,9,10,25,26)(3,13,19,12,14,31,15,30)(7,21,28,29,22,27,23,20);(2,9)(7,22)(8,10)(11,25)(20,29)(21,27)(23,28)(24,26)Polycyclic presentation:
2^9|d^2=fh,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gh,h^a=hi,e^b=ei,g^b=gi,f^c=fi,g^c=gh,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72213945, #72215833, … |
| maximal normal subgroup of | #72213937, #72213945, #72215833, … |
| maximal quotient of | #72224928, #72226796, #72231495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530344);