Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1435 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)(5,6,9,13,10,7,14,12)(8,11,16,19,17,15,20,18);(5,7)(6,10)(8,15)(9,12)(11,17)(13,14)(16,18)(19,20);(2,4)(8,16)(9,14)(11,18)(12,13)(15,19)(17,20);(5,8,9,16,10,17,14,20)(6,11,13,19,7,15,12,18)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224753, #72224755, #72224757 |
| maximal normal subgroup of | #72224753, #72224755, #72224757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288677);