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 | |
2353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,10)(8,15)(11,19)(12,16)(13,17)(18,20);(5,6)(7,11)(8,16)(9,12)(10,15)(13,18)(14,20)(17,19);(2,4)(7,13)(8,15)(10,16)(11,19)(14,17)(18,20);(5,7)(6,11)(8,17)(9,14)(10,18)(12,20)(13,15)(16,19)Polycyclic presentation:
2^9|e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338546, #72338554, #72338566 |
| maximal normal subgroup of | #72338546, #72338554, #72338566 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361187);