Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,10)(7,13)(8,15)(11,18)(12,16)(17,19);(5,6)(7,11)(8,16)(9,12)(10,15)(13,19)(14,18)(17,20);(2,4)(8,15)(14,20);(5,7,9,13)(6,11,12,19)(8,14,15,20)(10,17,16,18)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,h^b=hi,e^c=eh,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214565, #72219502, #72231758, … |
| maximal normal subgroup of | #72214565, #72219502, #72231758, … |
| maximal quotient of | #72214408, #72219501, #72231750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406125);