Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3280 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,25)(15,20,24,18)(16,21,17,26)(22,28,23,27);(6,9)(10,12)(13,15)(14,20)(16,22)(17,23)(18,25)(19,24)(21,28)(26,27);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(15,23)(16,27)(17,28)(18,20)(22,24)(25,26);(6,9)(10,12)Polycyclic presentation:
2^9|a^2=i,c^2=e,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=fh,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319285, #72319607, #72319654, … |
| maximal normal subgroup of | #72319285, #72319607, #72319654, … |
| maximal quotient of | #72322643, #72322834 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407319);