Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,10,11)(7,14,13,9)(8,16)(12,18)(15,17)(19,20);(1,2)(3,4)(5,7,9,6,10,13,14,11)(8,15,19,12,18,20,17,16);(2,4)(6,11)(8,17)(9,14)(12,20)(15,16)(18,19);(5,8)(6,12)(7,15)(9,19)(10,18)(11,16)(13,20)(14,17)Polycyclic presentation:
2^9|a^2=f,b^2=efi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319370, #72319399, #72319417, … |
| maximal normal subgroup of | #72319370, #72319399, #72319417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363838);