Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3722 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)(5,6,9,14,10,15,7,12)(8,13,16,20,17,11,18,19);(6,11)(8,17)(12,19)(13,15)(14,20)(16,18);(2,4);(5,7,10,9)(6,12,15,14)(8,16,17,18)(11,19,13,20)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,g^e=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261005, #72261190, #72261192, … |
| maximal normal subgroup of | #72261005, #72261190, #72261192, … |
| maximal quotient of | #72261738, #72279480, #72303499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400574);