Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,14,11,7,10,13)(8,12,17,20,18,15,19,16);(2,4)(5,7)(6,11)(8,16,18,20)(9,13)(10,14)(12,19,15,17);(1,2,3,4)(5,8)(6,12)(7,15)(9,17)(10,19)(11,18)(13,16)(14,20);(5,6)(7,11)(8,12)(9,14)(10,13)(15,18)(16,19)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72338538, #72338542, #72338559 |
| maximal normal subgroup of | #72338538, #72338542, #72338559 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274698);