Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480884 |
| 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 | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #37681 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#546 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1347 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)(7,15)(8,13)(9,11)(10,14,18,20)(12,16,19,17);(1,2)(3,4)(6,11)(7,8)(14,20)(16,17);(5,7,9,8)(6,12,11,19)(10,17,18,16)(13,20,15,14);(2,4)(5,8,9,7)(6,13,11,15)(10,17,18,16)(12,20,19,14)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285851, #72285863, #72286192, … |
| maximal normal subgroup of | #72285851, #72285863, #72286192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404560);