Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480866 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #39339 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#547 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11)(8,13)(9,17,18,12)(10,14)(15,19,16,20);(1,2)(3,4)(5,7,10,8)(6,11,14,13)(9,15,18,16)(12,20,17,19);(6,12)(7,8)(11,19)(13,20)(14,17)(15,16);(2,4)(5,8,10,7)(6,13,14,11)(9,16,18,15)(12,19,17,20)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,d^c=di,g^c=gh,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72312400, #72312403, #72312412, … |
| maximal normal subgroup of | #72312400, #72312403, #72312412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404493);