Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,16)(8,18)(9,14)(10,20)(11,13)(12,15)(17,19);(1,2)(3,4)(5,7)(6,9)(8,13)(10,18)(11,17)(12,14)(15,19)(16,20);(2,4)(6,13)(7,17)(14,18)(15,20);(2,4)(5,8,11,9)(6,14,13,18)(7,15,17,20)(10,19,12,16);(5,9)(6,7)(8,11)(10,19)(12,16)(13,17)(14,20)(15,18)Polycyclic presentation:
2^9|d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,e^d=ei,h^d=hi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213527, #72219565, #72219569, … |
| maximal normal subgroup of | #72213527, #72219565, #72219569 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421362);