Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4096 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)(6,11)(7,16)(8,9)(12,20)(14,15)(17,18);(2,4)(5,6)(7,13)(8,14)(9,11)(10,15)(12,16)(17,19)(18,20);(5,7)(6,12)(8,18)(9,16)(10,17)(11,19)(13,14)(15,20);(5,7)(6,13)(8,16)(9,18)(10,17)(11,20)(12,14)(15,19)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215961, #72219682, #72219683, … |
| maximal normal subgroup of | #72215961, #72219682, #72219683, … |
| maximal quotient of | #72226265, C2×#394127 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301323);