Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #36714 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20);(1,2)(3,4)(8,16)(12,18);(1,3,4,2)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(6,11)(10,17)(12,18)(13,19)Polycyclic presentation:
2^9|c^2=e,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72249443, #72249453, #72249466, … |
| maximal normal subgroup of | #72249443, #72249453, #72249466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406292);