Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500164 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24033 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61049 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,12)(4,17)(5,22)(6,21)(8,13)(9,23)(10,25)(14,15)(16,19)(18,20)(24,26)(27,31)(28,29)(30,32);(1,2)(3,13)(4,18)(5,23)(6,25)(7,11)(8,15)(9,22)(10,21)(12,27)(14,31)(16,29)(17,20)(19,30)(24,28)(26,32);(1,3,6,26,7,14,21,16)(2,8,10,30,11,27,25,28)(4,19,17,15,5,24,22,12)(9,29,23,31,20,32,18,13);(1,2)(3,8)(4,20)(5,9)(6,10)(7,11)(12,31)(13,15)(14,27)(16,28)(17,18)(19,32)(21,25)(22,23)(24,29)(26,30)Polycyclic presentation:
2^9|c^2=eh,f^2=hi,g^2=hi,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=ghi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72231700, #72231705, … |
| maximal normal subgroup of | #72213937, #72231700, #72231705, … |
| maximal quotient of | #72231523, C2×#500164, #72239854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407360);