Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500599 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,11)(7,15)(8,9)(12,16)(13,14)(17,20)(18,19);(1,2)(3,4)(5,6,8,11,10,14,9,13)(7,15,17,12,18,19,16,20);(2,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,15)(13,19)(14,20);(5,6,9,13,10,14,8,11)(7,12,17,19,18,20,16,15)Polycyclic presentation:
2^9|b^2=h,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214893, #72219588, #72219589, … |
| maximal normal subgroup of | #72214893, #72219588, #72219589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407796);