Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454039 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1685 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)(5,6,9,13,8,12,10,19)(7,14,17,20,16,11,18,15);(1,2,4,3)(5,7,8,16)(6,11,12,14)(9,17,10,18)(13,15,19,20);(2,3)(5,6)(7,15)(8,12)(9,19)(10,13)(11,17)(14,18)(16,20);(1,3,4,2)(5,6,8,12)(7,11,16,14)(9,13,10,19)(15,18,20,17)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285483, #72285522, #72285571, … |
| maximal normal subgroup of | #72285483, #72285522, #72285571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362343);