Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1644 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)(5,6,9,13,11,7,10,14)(8,16,17,12,18,20,19,15);(1,2,3,4)(5,7)(6,11)(8,12,18,15)(9,14)(10,13)(16,17,20,19);(2,4)(5,8)(6,12)(7,15)(9,17)(10,19)(11,18)(13,20)(14,16);(2,4)(6,13)(7,14)(9,10)(12,20)(15,16)(17,19)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72262977 |
| maximal normal subgroup of | #72262977 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340031);