Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1985 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)(5,6,8,12,10,14,9,13)(7,11,16,19,17,20,15,18);(2,4)(5,7,8,16,10,17,9,15)(6,11,12,19,14,20,13,18);(6,12,14,13)(7,15,17,16);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216067, #72219660, #72219668, … |
| maximal normal subgroup of | #72216067, #72219660, #72219668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275972);