Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453991 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,10)(8,17)(11,13)(12,18)(14,15)(19,20);(5,6)(7,11)(8,18)(9,12)(10,17)(13,15)(14,19)(16,20);(2,4)(5,6)(7,13)(8,18)(9,12)(10,17)(11,14)(15,20)(16,19);(5,7)(6,11)(8,15)(9,16)(10,19)(12,20)(13,18)(14,17)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361295);